# How do you graph log_a(b)?

See below

#### Explanation:

Usually with graphing questions, I ask the student to start with the basic graph and move forward from there. In this case, $y = \log x$ is a basic graph, so let's talk about how we'd find the shape.

Let's first remember that we're working with not just any log, but ${\log}_{10}$ - which is the default when the log function has no base showing.

Let's also remember that the relationship between log and ${10}^{x}$ is:

${\log}_{a} b = c \iff {a}^{c} = b$

a is our "starting number"
c is the number of times a is multiplied by itself
b is the result of the operation

In our graphing question, a=10 and we're graphing the interplay between b (the "x" value) and c (the "y" value). So let's do a quick table of values:

$\text{b"color(white)(00000) "c}$
$\text{1"color(white)(00000) "0}$
$\text{10"color(white)(0000) "1}$
$\text{100"color(white)(000) "2}$
$\frac{1}{10} \textcolor{w h i t e}{000} \text{-1}$
$\frac{1}{100} \textcolor{w h i t e}{00} \text{-2}$

So the basic pattern is that there is an asymptote that as b (our x value) approaches 0, c (our y value) heads for negative infinity and b increases towards infinity quickly as c grows slowly. Overall, the graph looks like this:

graph{logx [-1, 100, -5, 5]}