Technical Mathematics, Sixth Edition

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.

1. y = 2.48^3.50

2. 24.0 = b^3.50

3. 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,

1. y = 2.48^3.50 = 24.0

2. b = 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 102 = 100, we say that "2 is the logarithm of 100, to the base 10" because 2 is the exponent to which the base 10 must be raised to obtain 100. This is written log10 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 ...