### 18.2. Logarithms

Let us look at the exponential function

- y = b
^{x}

This equation contains three quantities. Given any two, we should be able to find the third.

▪ Exploration:

Try this. Find each missing quantity by calculator.

y = 2.48

^{3.50}24.0 = b

^{3.50}24.0 = 2.48

^{x}

Were you able to find y, b, and x?

TI-83/84 screen for the exploration.

We already have the tools to solve the first two of these. By calculator,

y = 2.48

^{3.50}= 24.0b = 24.0

^{1/3.50}= 2.48

However, none of the math we have learned so far will enable us to find x. For this and similar problems, we need logarithms.

#### 18.2.1. Definition of a Logarithm

The logarithm of some positive number y is the exponent to which a base b must be raised to obtain y.

## Example 12:Since 10 - log
_{10}100 = 2
Notice that the base is written as a subscript after the word log. The statement is read "The log of 100 (to the base 10) is 2." This means that 2 is an exponent because a logarithm is an exponent; 100 is the result of raising the base to that power. Therefore these two expressions are equivalent: |

Given the exponential function y = b^{x}, we say that "x is the logarithm of y, to the base b" because ...

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