Let us look at the exponential function
This equation contains three quantities. Given any two, we should be able to find the third.
Try this. Find each missing quantity by calculator.
y = 2.483.50
24.0 = b3.50
24.0 = 2.48x
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 = 2.483.50 = 24.0
b = 24.01/3.50 = 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.
The logarithm of some positive number y is the exponent to which a base b must be raised to obtain y.
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
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 = bx, we say that "x is the logarithm of y, to the base b" because ...