1105
y = log{base 10}(x) vs. x is inverse to y = 10^x vs. x. The tables are inverse, the graphs are inverse. You should be able to make a basic table and graph for both functions in 2 minutes or less.
Make a table of y = 10^x vs. x, then invert it to make a table of y = log{base 10}(x) vs. x. Sketch a quick graph of log{base 10}(x) vs. x.
What are log(100), log(.001) and log(10,000)?
log(100) = 2, easily read from the table. This happens because 10^2 = 100, which should be clear from looking at both tables.
Similarly, log(.001) = -3 because 10^-3 = .001.
log(10,000) = 4, easily figured by extending the table. This is so because 10^4 = 10,000.
Estimate log(50), log(-.3), log(.3) and log(7000).
50 lies between 10 and 100, so log(50) lies between log(10) = 1 and log(100) = 2. We might guess about halfway: maybe log(50) = 1.5. This is a pretty good guess.
Looking at the concavity of the graph we see that the actual value is probably a little higher, or about 1.7.
Use your calculator to find log(50), log(-.3) and log(7000).
What is 10^2.4? What is the log of this number?
What would you get if you took the log of both sides of the equation 10^(2x - 3) = .001?
We would get
log(10^2.4) = 2.4, log(10^aardvark) = aardvark, and log(10^(2x-3) ) = 2x - 3.
log(.001) = -3, since 10^-3 = .001.
So our equation becomes
Laws of exponents:
10^(a + b) = 10^a * 10^b
(10^a)^b = 10^(ab).
Laws of logarithms turn these laws around:
log(a * b) = log(a) + log(b)
log(a^b) = b * log(a).