Question

How do you graph `log_a(b)`?

Answer 1

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=logx` 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 <=> 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:

`"b"color(white)(00000) "c"`
`"1"color(white)(00000) "0"`
`"10"color(white)(0000) "1"`
`"100"color(white)(000) "2"`
`1/10color(white)(000) "-1"`
`1/100color(white)(00) "-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]}