Precalculus: Logarithmic Functions

Whereas an exponential function answers the question “A number raised to a power equals what?” a logarithmic function (or log function) answers the question “To what power must I raise a number to get another number?” In other words, the output for a logarithmic function is in actuality an exponent.

Specifically, the logarithmic expression log_c x (read “the log base c of x”) asks the question: c to what power equals x? Thus, the equations log_c x = n and cⁿ = x mean precisely the same thing.

Example 1: Find x in each of the equations.

(a)
log₃ 81 = x

This expression is the equivalent of 3^x = 81, so x = 4. It answers the question “3 to what power equals 81?”
(b)
log₂ x = −5

Rewrite as the equation 2⁻⁵ = x and evaluate; .
(c)
log_x 125 = 3

Rewrite as the equation x³ = 125 and take the cube root of each side; x = 5.
(d)
log_a 1 = x, where a is a positive integer

Rewrite as the equation a^x = 1. No matter the value of a, only one x value will result in a value of 1: x = 0, since any positive number raised to the 0 power is 1.

Natural and common logs

Although a logarithm's base can be any positive number (except for 1, since 1 raised to any real number will still be 1), there are two bases you'll encounter most often.

Base 10. A logarithm of base 10 is called a common log. In fact, if a logarithmic expression is written without specifying a base, that base is understood to be 10, in the same way that an unwritten exponent is understood to be 1.

Base e. Just like the exponential function with base e is called the natural exponential function, the logarithm with base e is called the natural logarithm. It is used so frequently that it has its own notation: ln x, and is read “the natural log of x” or “L-N of x,” in which case you actually say the letters L and N. Therefore, ln x is the same thing as log_e x.

Inverse relationship

Since exponential and logarithmic functions of the same base are inverses of one another, if you compose the two functions together, they will cancel one another out.

Since you will see common and natural logs most often, here is that inverse relationship expressed in terms of their respective bases:

Graphs of logarithmic functions

Since logarithmic and exponential functions are one another's inverses, it is easy to construct the graph of any logarithmic function y = log_a x based on the corresponding graph of y = a^x. Graphs of inverse functions are reflections of one another across the line y = x, since each graph contains the coordinates of the other graph, with each coordinate pair reversed. It is no surprise, then, that because all exponential graphs of the form y = a^x contain the point (0,1), then all logarithmic graphs of the form y = log_ax contain the point (1,0).

In Figure 1 , you can visually verify that the graphs of the natural logarithmic and natural exponential functions are, indeed, reflections of one another about the line y = x.

Figure 1

The graphs of y = e^x and y = in x are reflections of one another about the line y = x, as are all inverse functions.

Note that the domain of ln x, like all logarithmic functions of form y = log_ax, is (0,∞). Although it might appear that the y values of the logarithmic graph “level out,” as if approaching a horizontal asymptote, they do not. In fact, a logarithmic graph will grow infinitely tall, albeit much, much slower than its sister the exponential function. A range of (−∞,∞) for the logarithmic functions makes sense, since their inverses are exponential functions and have domains of (−∞,∞).

Change of base formula

With the aid of a scientific or graphing calculator, it is a simple matter to evaluate a logarithm. (It is not appropriate or necessary to learn to calculate complex decimal values of logarithms by hand.) However, you may notice that most computational tools have only two logarithmic buttons: one for common log and one for natural log. Thus, while it may be simple to calculate these values: