Refresher - Logarithmic Functions

Logarithmic Functions

A logarithmic function is one of the form

$f (x) = \log_{b} x$

where b, x > 0. Logarithmic functions are inverses of exponentials, so to find the graph of

$y = \log_{b} x$

, we can reflect the graph of $y = b^{x}$ in the line y = x.

In the previous section we studied the graph of an exponential function in the form of y = b^x. In this section we will study the inverse of the exponential function, the logarithmic function. Let's review the graph of $y = 2^{x}$ . To find the inverse of an exponential function we must reflect it over the line y = x. Recall from the section on inverses that we can find the equation of the inverse of a function algebraically by interchanging x and y, and solving for y.

The equation of the reflection of $y = 2^{x}$ is $x = 2^{y}$ . Since it is customary to write the equation that defines a function by expressing y in terms of x, we need to rewrite $x = 2^{y}$ so that it is solved for y. If we were to describe the equation $x = 2^{y}$ in words, we would say: “base 2 raised to the y power would give us x,” or “y is the exponent to base 2 needed to find x.” The word "logarithm” means exponent, so we can restate the sentence as: “y is the logarithm to the base 2 needed to find x.” In abbreviated form we would write: $y = \log_{2} x$

We can generalize and say, the equation $y = {log}_{b} (x)$ defines a logarithmic function that is the inverse of the exponential function y = b^x.

Characteristics of the graph of logarithmic functions:

The domain of $y = {log}_{b} (x)$ is all real positive numbers
The range of $y = {log}_{b} (x)$ is all real numbers
The curve approaches but does not intersect the y-axis. The y-axis is an asymptote of the curve
The graph of $y = {log}_{b} (x)$ is concave down

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Math 1A/1B: Pre-Calculus by Dr. Sarah Eichorn and Dr. Rachel Lehman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Links to an external site..