If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
017. `q uery 17
Question: `qquery Logarithms, Logarithmic Functions, Logarithmic Equations
1. For what value of x will the function y = log{base 2}(x) first reach 4?
Your solution:
Confidence Assessment:
Given Solution:
`a**
log{base 2}(x) = 4 translates to 2^4 = x
The value of x is therefore 2^4 = 16.
NEARLY-CORRECT STUDENT SOLUTION
log{base 2}(x) = 4 translates to 2^x = 4, and 2^2 = 4 so x = 2.
INSTRUCTOR RESPONSE
You have the right idea, but you confused the role of x when you translated log{base 2}(x) = 4.
As you will see below, the correct translation would be 2^4 = x.
It's very easy to get the variables reversed, so the translation has to be done carefully. It's best if you understand the reasons behind the translation, so the following explanations include the reasoning.
One solution:
The y = log{base b} function is inverse to
the y = b^x function.
This means that if you reverse
the columns of the y = b^x
table, you get the y = log{base b}(x) table.
When you reverse columns, you are reversing
the x and y variables.
For this reason, y = log{base b}(x) means the same thing as b^y = x.
Whether you completely understand about reversing the columns, etc., be sure you remember this.
You are asked to find the x value for which y = log{base 2}(x).
The value of y = log{base 2}(x) is y. So when the value is 4, this means that
4 = log{base 2}(x).
Now, since y = log{base b}(x) means the same thing as b^y = x , the equation 4 = log{base 2}(x) means the same thing as
2^4 = x.
Since 2^4 = 16, we conclude that x = 16.
Alternative solution:
We can make a table for y = log{base 2}(x)
y = log{base 2}(x) is defined at the inverse of the function of y = 2^x.
We start with a table for y = 2^x:
x y = 2^x -2 1/4 -1 1/2 0 1 1 2 2 4 3 8 4 16 5 32
We reverse the columns of the table, obtaining a table for y = log{base 2}(x)
x y = log{base 2}(x) 1/4 -2 1/2 -1 1 0 2 1 4 2 8 3 16 4 32 5 You should sketch rough graphs of these two functions.
From the tables, and from the graphs, it should be clear that y = log{base 2}(x) first takes value 4 when x = 16.
Self-critique (if necessary):
Self-critique rating:
Question: `qfor what value of x will the function y = ln(x) first reach y = 4?
Your solution:
Confidence Assessment:
Given Solution:
`ay = ln(x) means that e^y = x. The function y = ln(x) will first reach y = 4 when x = e^4 = 54.6 approx. **
ln(x) = 4 translates to x = e^4, which occurs at x = 55 approx.
ln(x) = 2 translates to x = e^2, which occurs at x = 7.4 approx.
ln(x) = 3 translates to x = e^3, which occurs at x = 20 approx. **
Self-critique (if necessary):
Self-critique rating:
Question: `q3. Explain why the y axis is an asymptote for a log{base b}(x) function
explain why the asymptote is the negative y axis if and only if b > 1
Your solution:
Confidence Assessment:
Given Solution:
`a** the log{base b} function is the inverse of the y = b^x function. Assuming b > 1, large negative values of x lead to
positive b^x values near zero, resulting in a horizontal asymptotes along the negative x axis.
When the columns are reversed we end up with small positive values of x associated with large negative y, resulting in a
vertical asymptote along the negative y axis.
You can take a negative power of any positive b, greater than 1 or not.
For b > 1, larger negative powers make the result smaller, and the negative x axis is an asymptote for the function y = b^x.
For b < 1, larger positive powers make the result smaller, and the positive x axis becomes the asymptote. **
STUDENT COMMENT
I understand it to read it but can’t remember it after I have read it.
INSTRUCTOR RESPONSE
Let's consider some of the statements in the given solution.
First statement: Assuming b > 1, large negative values of x lead to positive b^x values near zero, resulting in a horizontal asymptotes along the negative x axis.
Understanding what that means: Pick a value for b. This statement assumes b > 1, so let's pick something simple like b = 2. b = 3, or b = 10, or whatever, would also work, but we'll just use b = 2.
The statement says 'large negative values of x lead to positive b^x values near zero'. So let's test this for b = 2.
What happens to the value of 2^x for increasing negative values of x? For x = -1, -2, -3, -4 the values of b^x are 2^-1 = 1/2, 2^-2 = 1/4, 2^-3 = 1/8, 2^-4 = 1/16.
Now 1/16 is clearly positive, and clearly pretty close to zero. And we aren't even into very large negative values of x. If we continue to x = -5, -6, -7, ..., our y = 2^x values stay positive, but keep getting closer and closer to 0. By the time x = -10, our y value is 1/1024, positive but less that .001.
There's no limit to how close our values can get to zero. However they always stay positive.
Now if you plot y vs. x for x = -1, -2, -3 and -4, you will see how the asymptote forms. As you move to the left along the negative x axis, the y values keep getting smaller and smaller. The graph continues to approach, but never reaches, the negative x axis.
A table of values for y = 2^x:
x y = 2^x -4 1/16 -3 1/8 -2 1/4 -1 1/2 0 1 1 2 We chose b = 2 as a basis for this discussion. You might want to see what happens if b = 3, and if b = 10. You will find that the y values again stay positive but as we move through x values -1, -2, -3, -4, ..., we approach zero much more quickly than for b = 2. We still have an asymptote at the negative x axis, but the graph approaches it much more quickly for larger values of b.
Second statement: When the columns are reversed we end up with small positive values of x associated with large negative y, resulting in a vertical asymptote along the negative y axis.
Understanding what this means:
We are talking about reversing the columns of our y = b^x table.
For our previous b = 2 example, the reversed table is the function y = log{base b}(x), for b = 2:
x y = log{base 2}(x) 1/16 -4 1/8 -3 1/4 -2 1/2 -1 1 0 2 1 Graph this. You will see how the asymptote forms with the negative y axis.
Statement 3: For b < 1, larger positive powers make the result smaller, and the positive x axis becomes the asymptote.
Understanding what this means: Pick a positive value of b which is less than 1. Here we'll pick b = 1/2.
What happens to the function y = (1/2)^x for larger and larger positive values of x?
For x = 1, 2, 3, 4, ..., we find that y = (1/2)^x takes values 1/2, 1/4, 1/8, 1/16, ... .
These are the same values obtained in the preceding example y = 2^x for x = -1, -2, -3, -4, ... .
If you make a table and a graph you will see that the y = (1/2)^x function approaches an asymptote with the positive x axis.
Additional insights:
The same thing will happen for any value of b which is less than 1. If b is less than 1, every multiplication by b gives a smaller result than before, so the values of y = b^x continue to decrease as x increases.
If you reverse the columns of the y = (1/2)^x table, you get the y = log{base 1/2)(x) table. If you graph this table, you will see how the y = (1/2)^x asymptote with the positive x axis becomes an asymptote of the log function with the positive y axis.
Self-critique (if necessary):
Self-critique rating:
Question: `q5. What are your estimates for the values of b for the two exponential functions on the given graph?
Your solution:
Confidence Assessment:
Given Solution:
`a** The exponential function y = A b^x has y intercept (0, A) and basic point (1, A * b).
Both graphs pass thru (0, 1) so A = 1.
The x = 1 points are (1, 3.5) and (1, 7.3) approx.. Thus for the first A * b = 3.5; since A = 1 we have b = 3.5. For the
second we similarly conclude that b = 7.3.
So the functions are y = 3.5^x and y = 7.3^x, approx.. **
Self-critique (if necessary):
Self-critique rating:
Question: `qAt what points will each of the logarithmic functions reach the values y = 2, y = 3 and y = 4?
Your solution:
Confidence Assessment:
Given Solution:
`a** Using trial and error we find that for y = 3.5^x, the x values .55, .88 and 1.11 give us y = 2, 3 and 4.
We find that for y = 7.3^x, the x values .35, .55 and .70 give us y = 2, 3 and 4. **
Self-critique (if necessary):
Self-critique rating:
Question: `q7. What is the decibel level of a sound which is 10,000 times as loud as hearing threshold intensity?
Your solution:
Confidence Assessment:
Given Solution:
`adB = 10 log(I / I0). If the sound is 10,000 times as loud as hearing threshold intensity then I / I0 = 10,000.
log(10,000) = 4, since 10^4 = 10,000.
So dB = 10 log(I / I0) = 10 * 4 = 40.
STUDENT QUESTION
I didnt understand were log(10000) is 4 came from i think i would understand
the problem if i new how to come up with that
INSTRUCTOR RESPONSE
The reason is in the line
'log(10,000) = 4, since 10^4 = 10,000'
Recall that log(x) and 10^x are inverse functions.
So 10^4 = 10 000 means the same thing as log(10 000) = 4.
The formal definition is in terms of inverse functions. You can also think
of the log of a number as the power to which 10 must be raised to get the
number.
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat are the decibel levels of sounds which are 100, 10,000,000 and 1,000,000,000 times louder than threshold intensity?
Your solution:
Confidence Assessment:
Given Solution:
`a10 log(100) = 10 * 2 = 20, so a sound which is 100 times hearing threshold intensity is a 20 dB sound.
10 log(10,000,000) = 10 * 7 = 70, so a sound which is 100 times hearing threshold intensity is a 70 dB sound.
10 log(1,000,000,000) = 10 * 9 = 90, so a sound which is 100 times hearing threshold intensity is a 90 dB sound.
Self-critique (if necessary):
Self-critique rating:
Question: `qhow can you easily find these decibel levels without using a calculator?
Your solution:
Confidence Assessment:
Given Solution:
`aSince 10^1, 10^2, 10^3, ... are 10, 100, 1000, ..., the power of 10 is the number of zeros in the result. Since the log
of a number is the power to which 10 must be raised to get this number, the log of one of these numbers is equal to the
number of zeros.
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat are the decibel levels of sounds which are 500, 30,000,000 and 7,000,000,000 times louder than threshold intensity?
Your solution:
Confidence Assessment:
Given Solution:
`a10 log(500) = 10 * 2.699 = 26.99 so this is a 26.99 dB sound.
10 log(30,000,000) = 10 * 7.477 = 74.77 so this is a 74.77 dB sound.
10 log(7,000,000,000) = 10 * 9.845 = 98.45 so this is a 98.45 dB sound.
Self-critique (if necessary):
Self-critique rating:
Question: `q8. If a sound measures 40 decibels, then what is the intensity of the sound, as a multiple of the hearing threshold intensity?
Your solution:
Confidence Assessment:
Given Solution:
`a** Let x be the ratio I / I0. Then we solve the equation 40= 10*log(x). You solve this by dividing by 10 to get
log(x) = 4 then translating this to exponential form
x = 10^4 = 10,000.
The sound is 10,000 times the hearing threshold intensity, so
I = 10,000 I0. **
Self-critique (if necessary):
Self-critique rating:
Question: `qAnswer the same question for sounds measuring 20, 50, 80 and 100 decibels.
Your solution:
Confidence Assessment:
Given Solution:
`a** since dB = 10 log(I / I0) we have
log(I/I0) = dB / 10. Translating to exponential form this tells us that
I / I0 = 10^(dB/10) wo that
I = I0 * 10^(dB/10).
For a 20 dB sound this gives us
I = I0 * 10^(20/10) = I0 * 10^2 = 100 I0. The sound is 100 times the intensity of the hearing threshold sound.
For a 50 dB sound this gives us
I = I0 * 10^(50/10) = I0 * 10^5 = 100,000 I0. The sound is 100,000 times the intensity of the hearing threshold sound.
For an 80 dB sound this gives us
}I = I0 * 10^(80/10) = I0 * 10^8 = 100,000,000 I0. The sound is 100,000,000 times the intensity of the hearing
threshold sound.
For a 100 dB sound this gives us
I = I0 * 10^(100/10) = I0 * 10^10 = 10,000,000,000 I0. The sound is 10,000,000,000 times the intensity of the
hearing threshold sound. **
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat equation you would solve to find the intensity for decibel levels of 35, 83 and 117 dB.
Your solution:
Confidence Assessment:
Given Solution:
`a** the equation to find I is dB = 10 log(I / I0) so the equations would be
35 = 10 log(I / I0)
83 = 10 log(I / I0)
117 = 10 log(I / I0).
The solution for I in the equation dB = 10 log(I / I0) is
I = I0 * 10^(dB/10). For the given values we would get solutions
10^(35/10) I0 = 3162.3 I0
10^(83/10) I0 = 199526231.5 I0
10^(117/10) I0 = 501187233627 I0 **
Self-critique (if necessary):
Self-critique rating:
Question: `q9. is log(x^y) = x log(y) valid? If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`a** log(a^b) = b log a so log(x^y) should be y log (x), not x log(y). **
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x/y) = log(x) - log(y) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aYes , this is valid. It is the inverse of the exponential law a^x / a^y = a^(x-y).
STUDENT COMMENT
I can verify these just fine if I have a copy of the laws in front of me; but I don't see any way to remember them all.
INSTRUCTOR RESPONSE
Let's look at a way to understand the laws.
log( x * y ) = log(x) + log(y).
For example if x = 1000 and y = 100:
x * y = 1000 * 100 = 10^3 * 10^2 = 10^(3 + 2) = 10^5 so log(x * y) = log(10^5) = 5.
log(x) = log(100) = log(10^2) = 2, so log(x) = 2.
log(y) = log(1000) = log(10^3) = 3, so log(y) = 3.
log(x) + log(y) = 2 * 3 = 5, and log(x * y) = 5.
So for this example we have verified that log(x * y) = log(x) + log(y).
If we think about it we also see why this must be true. When we multiply x * y the exponents of these numbers add. Since the log is the exponent, the logs therefore add.
log(x / y) = log(x) - log(y).
For example if x = 1000 and y = 100:
x / y = 1000 / 100 = 10^3 / 10^2 = 10^(3 - 2) = 10^1, so log(x / y) = log(10^1) = 1.
log(x) - log(y) = 3 - 2 = 1.
This verifies the law, and shows how it is connected to the law of exponents for division.
log(x^a) = a log(x).
For example if x = 1000 and a = 5:
log(x^a) = log(1000 ^ 5) = log( (10^3)^5 ) = log(10^15) = 15.
a log(x) = 5 * log(10^3) = 5 * 3 = 15.
This verifies the law and shows how it is connected to the law of exponents for powers of a number.
Self-critique (if necessary):
Self-critique rating:
Question: `qis log (x * y) = log(x) * log(y) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aNo. log(x * y) = log(x) + log(y)
Self-critique (if necessary):
Self-critique rating:
Question: `qis 2 log(x) = log(2x) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`a** log(a^b) = b log a so 2 log(x) = log(x^2), not log(2x). **
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x + y) = log(x) + log(y) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`a** log(x) + log(y) = log(xy), not log(x+y). **
STUDENT QUESTION:
I understand that log(x) + log(y) isn't log(x + y), because the rule says that log(x) + log(y) = log(x y).
Still it looks like log(x) + log(x) should be log(x + y), and I'm having trouble keeping this straight.
INSTRUCTOR RESPONSE:
Another way to see that it's not so is to use numbers.
log(1000) = 3 and log(100) = 2, so log(1000) + log(100) = 3 + 2 = 5.
However log(1000 + 100) = log(1100). Since 1100 is between 10^3 = 1000 and 10^4 = 10 000, log(1100) is between 3 and 4. (Also since 1100 is a whole lot closer to 1000 than to 10 000, we know that log(1100) is a lot closer to 3 than to 4).
In any case, log(1000 + 100) certainly isn't 5.
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x) + log(y) = log(xy) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aThis is value. It is inverse to the law of exponents a^x*a^y = a^(x+y)
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x^y) = (log(x)) ^ y valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aNo. log(x^y) = y log (x).
This is the invers of the law (x^a)^b = x^(ab)
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x - y) = log(x) - log(y) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aNo. log(x)-log(y) = log x/ log y, not log(x-y).
Self-critique (if necessary):
Self-critique rating:
Question: `qis 3 log(x) = log(x^3) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aYes. log(x^a) = a log(x).
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x^y) = y + log(x) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aNo. log(x^y) = y log(x).
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x/y) = log(x) / log(y) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aNo. log(x/y) = log(x) - log(y).
Self-critique (if necessary):
Self-critique rating:
Question: `qis log(x^y) = y log(x) valid. If so why, and if not why not?
Your solution:
Confidence Assessment:
Given Solution:
`aThis is valid.
Self-critique (if necessary):
Self-critique rating:
Question: `q10. what do you get when you simplify log {base 8} (1024)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Given Solution:
`aCOMMON ERROR: log {base 8} (1024) = Log (1024) / Log (8) = 3.33333
EXPLANATION:
log {base 8} (1024) = Log (1024) / Log (8) is correct, but 3.33333 is not an exact answer.
log {base 8 } (1024) = log {base 8 } (2^10).
Since 8 = 2^3, 2^10 = 2^(3 * 10/3) = (2^3)^(10/3) = 8^(10/3).
Thus log {base 8} 1024 = log{base 8} 8^(10/3) = 10/3.
Note that 10/3 is not exactly equal to 3.33333. You need to give exact answers where possible.
STUDENT QUESTION
I recognized from computer terms that 8 and 1024 were related, but I wasn't
sure how they
could be brought to the same level so to speak for the logs to be evaluated
into fractions. I think this is the confusing bit
for me: 2^10 = 2^(3 * 10/3). I see that 3*10/3 = 10, but I'm not sure how
you arrived at making this relationship.
Could you provide a little more explanation?
INSTRUCTOR RESPONSE
We want to find what power of 8 is equal to 1024. We know that 8 = 2^3 and
1024 = 2^10.
So the question becomes:
what power of 2^3 is 2^10?
We can reason out the answer as indicated, but I would agree if you said
that isn't something that will naturally occur to most students.
Alternatively we can let p be the power we're looking for and write
(2^3)^p = 2^10.
We easily enough solve this by applying the laws of exponents. Since (2^3)^p
= 2^(3 p) we get
2^(3 p) = 2^10 so that
3 p = 10 and
p = 10/3.
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat do you get when you simplify log {base 2} (4 * 32)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Given Solution:
`a** log{base 2}(4*32) = log{bse 2}(2^2 * 2^5) = log{base 2}(2^7) = 7. **
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat do you get when you simplify log (1000)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Given Solution:
`aSince 10^3 = 1000, we have log (1000) = 3
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat do you get when you simplify ln(3xy)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Given Solution:
`a ln(3xy) = ln (3) + ln(x) + ln(y) = 1.0986 + ln(x) + ln(y)
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat do you get when you simplify log(3) + log(7) + log(41)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Given Solution:
`a log(3) + log(7) + log(41) = log (3*7*41). 3 * 7 * 41 is not a rational-number power of 10 so this can't be evaluated
exactly.
Self-critique (if necessary):
Self-critique rating:
Question: `q11. Show how you used the given values to find the logarithm of 12. Explain why the given values don't help much if you want the log of 17.
Your solution:
Confidence Assessment:
Given Solution:
`a** The problem was as follows: Given that log(2) = .301, log(3) = .477, log(5) = .699, log(7) = .845, find log(4),
log(6), log(8), and log(9). Use the given values to find the logarithm of every possible integer between 11 and 20.
To get log(12), given the logs of 2, 3, 5 and 7, you have to break 12 down into a product of these numbers.
Since 12 = 2 * 2 * 3 we have
log(12) = log(2) + log(2) + log(3) = .301 + .301 + .477 = 1.079.
Your calculator will confirm this result. **
Self-critique (if necessary):
Self-critique rating:
Question: `q12. What do you get when you solve 3 ^ (2x) = 7 ^ (x-4), and how did you solve the equation?
Your solution:
Confidence Assessment:
Given Solution:
`a ** log[3^(2x)]= log [7^(x-4)]. Using the laws of logarithms we get
2xlog(3)= (x-4) log(7). The distributive law gives us
2xlog(3)= xlog(7)- 4log(7). Rearranging to get all x terms on one side we get
2xlog(3)- xlog(7)= -4log(7). Factor x out of the left-hand side to get
x ( 2 log(3) - log(7) ) = -4 log(7) so that
x = -4 log(7) / [ 2 log(3) - log(7) ].
Evaluating this we get x = -31, approx. **
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat do you get when you solve 2^(3x) + 2^(4x) = 9, and how did you solve the equation?
Your solution:
Confidence Assessment:
Given Solution:
`aCOMMON ERROR: 3xlog(2) + 4xlog(2) = 9
Explanation:
Your equation would require that log( 2^(3x) + 2^(4x) ) = log(2^(3x) ) + log(2^(4x)).
This isn't the case. log(a + b) is not equal to log(a) + log(b). log(a) + log(b) = log(a * b), not log(a + b).
If this step was valid you would have a good solution.
However it turns out that this equation cannot be solved exactly for x. The best we can do is certain sophisticated forms
of trial and error. **
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat do you get when you solve 3^(2x-1) * 3^(3x+2) = 12, and how did you solve the equation?
Your solution:
Confidence Assessment:
Given Solution:
`a** 3^(2x-1) * 3^(3x+2) = 12. Take log of both sides:
log{3} [3^(2x-1) * 3^(3x+2)] = log{3} 12. Use log(a*b) = log(a) + log(b):
log{3}(3^(2x-1)) + log{3}(3^(3x+2) = log{3} 12. Use laws of logs:
(2x-1) + (3x+2) = log{3} 12. Rearrange the left-hand side:
5x + 1 = log{3}12. Subtract 1 from both sides then divide both sides by 5:
x = (log {3}(12) -1)/ 5. Use the fact that log{b}(x) = log x / (log b) to get
x = (log(12) / log(3) - 1) / 5. Evaluate using calculator:
x = .2524 **
Alternate solution:
log[3^(2x-1) * 3^(3x+2)] = log (12)
<h3>Starting with
3^(2x-1) * 3^(3x+2) = 12, take the log of both sides to get
log(3^(2x-1) * 3^(3x + 2) ) = log(12). Using log(xy) = log x + log y we get
log(3^(2x-1)) + log( 3^(3x + 2) ) = log 12. Using log(x^a) = a log x we get
(2x-1) log 3 + (3x+2) log 3 = log 12. Factor log 3 on the left to get
(2x - 1 + 3x + 2) log 3 = log 12 so that
5x + 1 = log(12) / log(3). Solve for x, obtaining
x = ( log(12) / log(3) - 1) / 5, as before. Evaluate to get
x = .2524, approx..
Self-critique (if necessary):
Self-critique rating:
Question: `qquery fitting exponential functions to data
1. what is the exponential function of form A (2^(k t) ) such that the graph passes thru points (-4,3) and (7,2), and what
equations did you solve to obtain your result?
Your solution:
Confidence Assessment:
Given Solution:
`a** Recall when we substituted the coordinates of three points into the form y = a x^2 + b x + c of a quadratic, then solved the resulting system of three linear equations to get the values of a, b and c. We do something similar here, but this time the form is y = A * 2^(kx). Our two unknowns will be A and k, and we will only require two equations to solve for our two unknowns. We therefore need only two points.
Substituting data points into the form y = A * 2^(kx) we get
3 = A * 2^(-4k) (from the data point (-4,3)) and
2 = A * 2^( 7k) (from the data point (7,2)).
When we solved the three equations for the quadratic model, we added multiples of equations in order to eliminate variables. However adding multiples of equations doesn't work here. What does work is dividing one equation by another, which eliminates the variable A.
Dividing the first equation by the second we get
3/2 = A * 2^(-4k) / (A * 2^(7k) ), which simplifies to
1.5 = 2^(-4k)/ 2^(7k).
Now by the laws of exponents 2^(-4k)/ 2^(7k) = 2^(-4-7k)= 2^(-11 k), so that
log(2^(-11 k)) = log(1.5) and
-11 k * log(2) = log 1.5. We solve this for k, obtaining
k = log(1.5) / (-11 log(2)).
Evaluating with a calculator we find that k = -.053, approx..
Just as we did when solving for the quadratic model, once we have evaluated an unknown we substitute that value into our original equations.
Our first equation was 3 = A * 2^(-4k), which is easily solve for A to give us
A = 3 / (2 ^(-4k) ).
Substituting k = -.053 we get
A= 3/ 1.158 = 2.591.
Now we know the values of both A and k, which we substitute into our form y = A * 2^(kx) to obtain our model
y = 2.591(2^-.053t). **
STUDENT COMMENT:
I'm trying, but I just don't understand the given solution.
INSTRUCTOR RESPONSE:
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat is the exponential function of form A e^(k2 t) such that the graph passes thru points thru points (-4,3) and (7,2) and how did you solve the equations to find this function?
Your solution:
Confidence Assessment:
Given Solution:
`a** Substituting data points into the form y = A * e^(kx) we get
3= A * e^(-4k) and
2= A * e^(7k)
Dividing the first equation by the second we get
1.5= e^(-4k)/ e^(7k)= e^(-4-7k)= 2^(-11k)
so that
ln(e^(-11k)) = ln(1.5) and
-11 k = 1.5 so that
k= ln(1.5) / (-11). Evaluating with a calculator:
k= -.037 approx.
From the first equation
A = 3 / (e ^(-4k) ). Substituting k = -.037 we get
A= 3/ 1.158 = 2.591.
So our form y = A * e^(kx) gives us
y= 2.591(e^-.039 t). **
Self-critique (if necessary):
Self-critique rating:
Question: `qwhat is the exponential function of form A b^t such that the graph passes thru points thru points (-4,3) and (7,2).
Your solution:
Confidence Assessment:
Given Solution:
`a** Our equations are
3= Ab^-4
2= Ab^7
3/2= Ab^-4/(Ab^7)
1.5= b^-11
b= .96
3= A * .96 ^ -4
3= A * 1.177
2.549= A
y= 2.549 * .96^t **
Self-critique (if necessary):
Self-critique rating:
Question: `q2. Find the exponential function corresponding to the points (5,3) and (10,2).
Your solution:
Confidence Assessment:
Given Solution:
`a** Using y = A b^t we get equations
3= Ab^5
2= Ab^10
Dividing first by second:
3/2= Ab^5/(Ab^10).
1.5= b^-5
b= .922
Now A = 3 / b^5 = 3 / .922^5 = 4.5.
Our model is
y = 4.5 * .922^t. **
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat are k1 and k2 such that b = e^k2 = 2^k1?
Your solution:
Confidence Assessment:
Given Solution:
`a** .922 = e^k2 is directly solved by taking the natural log of both sides to get
k2 = ln(.922) = -.081.
.922= 2^k1 is solved as follows:
log(.922) = log(2) k1
k1 = log(.922) / log(2) = -.117 approx..
Using these values for k1 and k2 we get
}g(x) = A * 2^(k1 t) = 4.5 * 2^(-.117 t) and
h(x) = A e^(k2 t) = 4.5 e^(-.081 t). ****
STUDENT QUESTION
I understand the process but I'm a little lost as to what b = e^k2 = 2^k1 is
supposed to tell
us. What are k1 and k2 and how are they related?
INSTRUCTOR RESPONSE
The form A e^(kx) is pretty standard for exponential functions. The main
reason is that it's very easy to do calculus when the base is e; that's the
most natural base to use for calculating rate-of-change functions; and
that's why its inverse function is called the 'natural log').
The form A * 2^(kx) is useful because in this form we can easily calculate
doubling time and/or halflife (using this form it's not hard to see why
doubling time is the value of x such that k x = 1, halflife is the value of
x such that kx = -1).
To get the relationship between k_2 and k_1:
Starting with e^k_2 = 2^k_1 , take the natural log of both sides to get
k_2 = ln(2^(k_1)) = k_1 ln(2).
Thus k_2 = k_1 ln(2), and that's the relationship between k_1 and k_2.
Self-critique (if necessary):
Self-critique rating:
Question: `q3. earthquakes measure R1 = 7.4 and R2 = 8.2.
What is the ratio I2 / I1 of intensity and how did you find it?
Your solution:
Confidence Assessment:
Given Solution:
`a** R1 = log(I1 / I0) and R2 = log(I2 / I0) so
I1/I0 = 10^R1 and I1 = 10^R1 * I0 and
I2/I0 = 10^R2 and I2 = 10^R2 * I0 so
I2 / I1 = (I0 * 10^R2) / (I0 * 10^R1) = 10^R2 / 10^R1 = 10^(R2-R1).
So if R2 = 8.2 and R1 = 7.4 we have
I2 / I1 = 10^(R2 - R1) = 10^(8.2 - 7.4) = 10^.8 = 6.3 approx.
An earthquake with R = 8.2 is about 6.3 times as intense as an earthquake with R = 7.4. **
Self-critique (if necessary):
Self-critique rating:
Question: `q4. I2 / I1 ratios
If one earthquake as an R value 1.6 higher than another, what is the ratio I2 / I1?
Your solution:
Confidence Assessment:
Given Solution:
`a** As before I2 / I1 = 10^(R2-R1). If R2 is 1.6 greater than R1 we have R2 - R1 = 1.6 and
I2 / I1 = 10^1.6 = 40 approx.
An earthquake with R value 1.6 higher than another is 40 times as intense. **
Self-critique (if necessary):
Self-critique rating:
Question: `qIf one earthquake as an R value `dR higher than another, what is the ratio I2 / I1?
Your solution:
Confidence Assessment:
Given Solution:
`a** As before I2 / I1 = 10^(R2-R1). If R2 is `dR greater than R1 we have R2 - R1 = `dR and
I2 / I1 = 10^`dR. **
Self-critique (if necessary):
Self-critique rating:
If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.
Question: For what value of x will the value of the function y = log{base 3}(x) first reach 4?
Your solution:
Confidence Assessment:
Question: Explain why the negative y axis is an asymptote for the function y = log{base 2}(x).
Your solution:
Confidence Assessment:
Question: How can you determine, without the use of a calculator, the decibel level of a sound whose intensity is 10 000 times that of the hearing threshold?
Your solution:
Question: Is log(x^y) = x ^ log(y) valid? If so why, and if not why not?
Your solution:
Confidence Assessment:
Question: Is log(x*y) = log(x) + log(y) valid? If so why, and if not why not?
Your solution:
Confidence Assessment:
Question: What do you get when you simplify log {base 8} (256)? If it can be evaluated exactly, what is the result and how did you get it?
Your solution:
Confidence Assessment:
Question: Solve the equation 7^(2x) = 3^(x+2).
Your solution:
Confidence Assessment:
Question: What is the exponential function of the form y = A b^x which passes through the points (4, 5) and (6, 125)?
Your solution:
Self-critique rating: