Logarithms

For simplicity, we restrict our attention to common logarithms, though any logarithm will work for these calculations.

First, the definition. For any positive number X, the logarithm of X, log(X), is the number making

10log(X) = X

true. That is, log(X) and 10X are inverses of one another. For example,

log(10) = 1 because 101 = 10 log(100) = 2 because 102 = 100 log(1) = 0 because 100 = 1 log(17.5) = 1.24304 because 101.24304 = 17.5

By its design, the logarithm is a tool for extracting exponents.

We will need to know three things about the logarithm.

(1) How to compute it. (Use the log button on a calculator.) (2) For all positive numbers X and Y, log(X⋅Y) = log(X) + log(Y). (2) For all positive numbers X and all numbers Y, log(XY) = Y⋅log(X).

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