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Logarithms

Why Use Logarithms?

Suppose we have a very efficient way of writing any number as powers of a fixed number, say 10:

x = 10^m

y = 10ⁿ

Then we can perform multiplication and division by using addition and subtraction (of exponents):

xy = 10^m+n

= 10^m−n

A logarithm is just another name for an exponent with respect to a certain base such as 10.

The General Logarithm

Let b > 0.

If  y = b^x,    then we write log_b(y) = x or simply  log_b y = x    (read "log of y with base b")

Note:   log_b y is not defined for y ≤ 0 , as y = b^x > 0 for any x (as b > 0).

Example

More Examples

Exercise

For this exercise, evaluate the logarithm, to the base shown, of the number.