Graphing a Log Function

(Video) Graphing Logarithmic Functions

by ThinkwellVids

Here's a video by ThinkwellVids going over the graph of logarithms. In the video, he explains that to graph these functions the best approach is to pick some numbers and plug them in. He shows that the larger the base, the slower the graph grows along the $y$-axis.

(Video) Matching Logarithmic Functions with Their Graphs

by ThinkwellVids

Here's another ThinkwellVids video going over some exercises to draw a graph given a logarithmic function. There are different ways that the standard logarithmic graph can be changed. $y=\log{x\pm a}$ shifts the graph left or right depending on if it's plus or minus respectively. $y=\log{x}\pm a$ shifts the graph up or down depending on if it's plus or minus respectively. $y=\log{ax}$ stretches or compresses depending on if $a>1$ or $0<a<1$ respectively.

Graphing a Log Function

Graphing logarithmic functions (with real numbers) is kind of different from graphing other types of functions. The best way to graph logarithmic functions is to graph exponential functions, then find the inverse of that graph.

So, our first step will be to graph the exponential function, $a^x$. Once we've done that, we want to reflect the exponential function over the line $y=x$, which will give the inverse function. This new inverse function is our logarithmic function.

Steps:

1. Graph the exponential function.

2. Draw $y=x$.

3. Reflect the function over $y=x$.

Example:

Graph the function $\log_3 (x)$.

Step 1: Graph the exponential function.

The corresponding exponential function for $\log_3 (x)$ is $3^x$. The graph of $3^x$ looks like:

Step 2: Draw $y=x$.

Step 3: Reflect the function over $y=x$.

We have graphed our logarithmic function! Here is a graph of $\log_3 (x)$ on its own.

Looking Forward

Log functions will have certain features, like horizontal and vertical asymptotes, which you will learn about in future lessons. You'll also learn how to add and subtract log functions with the same base.

Graphing a Log Function

A logarithmic function is the opposite of an exponential function.

The most basic log function is: $$y=\log_b (x)$$

We verbally say, "log base $b$ of $x$". That is exactly what this function is: a logarithm with base $b$ of $x$. It tells us to what power we raise $b$ in order to get $x$.

To graph a log function, it helps to make a table of values. Choose $x$ values that are powers of the base, then solve for $y$. For example, if $b=10$,

• $\log_{10}(1)=0$
• $\log_{10}(10)=1$
• $\log_{10}(100)=2$

Plot those points on the graph. You'll notice that the graph passes through the point $(0,1)$ and rises slowly as $x$ increases.

Log graphs will always have a vertical asymptote at $x=0$ because you can't take the log of $0$ or a negative number. The graph gets very close to the $y$-axis but never touches it.

If you're curious, you can also learn how to find the range and domain algebraically.