How do you graph #log_a(b)#?

1 Answer

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]}