3.8 Graphs of Logarithmic Functions

Let’s take a basic logarithmic function such that

$f(x)={\log }_{a}x$

where $a>1$. There are three properties of a graph of the function for any $a$ greater than 1. First, $f(x=1)=0$. Second, the logarithmic function is increasing with $x$. Therefore, as the value of $x$ increases, the outcome of the function increases. Lastly, $f(x)$ is an inverse of $x=a^{\mathit{\text{f}}(x)}$. The graph of the logarithmic function looks like this:

When $0<a<1$, the graph of $f(x)$ still goes through the same x-intercept (1, 0), but it is a decreasing function.

In addition, if a positive real number $b$ is added within the logarithm, it will shift the graph left by $b$. If a positive real number $b$ is subtracted within the logarithm, it will shift the graph right by $b$. If a positive real number $c$ is added to the entire function, it will shift the graph up by $c$. If a positive real number $c$ is subtracted from the entire function, it will shift the graph down by $c$.

Let’s graph the following two functions.

$f(x)=\ln (x-3)+2$

$g(x)=\ln (x+1)-1$

Let’s start with $f(x)$. It is moved to the right by 3 and up by 2. For the function $g(x)$, the graph is moved to the left by 1 and up by 1. They should look like this: