1110

• Run Multiple Choice Generator at h:\shares\physics
• Run Core Problems by giving the program 174 as your 'course'--enter this when asked for your 3-digit course number.  
• Do the 'quizzes' on Exponential Functions: Numbers and Exponential Functions: Graphs.
• If you miss a problem be sure to click on Comment and explain what you do and do not understand about the question and the given solution

• Solve the equation log(y) = 5 log(x) + 7 for y.  What sort of function is y(x)?

Starting with

• log(y) = 5 log(x) + 7 we take apply the inverse function 10^x to both sides to obtain
• 10^log(y) = 10^(5 log(x) + 7). 

The left-hand side is just y; using the laws of exponents on the right-hand side we get

• y = 10^(5 log(x) ) * 10^7. 

Since 10^(ab) = (10^a)^b the factor 10^(5 log(x)) = (10^(log(x))^5) so

• y = ( 10^log(x) ) ^ 5 * 10^7.  Since 10^log(x) = x we have
• y = 10^7 * x^5.

So y is a power function of form y = A x^p with A = 10^7 and p = 5.

• Solve the equation log(y) = 5 x + 7.  What sort of function is y(x)?

Starting with

• log(y) = 5 x + 7 we take apply the inverse function 10^x to both sides to obtain
• 10^(log(y)) )= 10^(5x + 7).

The left-hand side is just y; using the laws of exponents on the right-hand side we get

• y = 10^(5x) * 10^7.

Since 10^(ab) = (10^a)^b the factor 10^(5 x) = (10^5)^x so

• y = 10^7 * (10^5)^x = 10^7 * 100,000^x.

This function is an exponential function of form y = A b^x with A = 10^7 and b = 100,000.

Generalizing:

If

• log(y) = m log(x) + b

then when we solve for y we get a power function.

If

• log(y) = m x + b

then when we solve for y we get an exponential function.

If we have a table of y vs. x data, then if a graph of log(y) vs. log(x) gives us a straight line, we can find the slope m and the y-intercept b and get the equation log(y) = m log(x) + b.

Of a graph of log(y) vs. x is linear, then the slope and y intercept will give us the form log(y) = m x + b.