Question

How do you show that the slope of `a^y=b^x` is `log_a b`?

Answer 1

I used some properties of logs:

Explanation:

Let us try to isolate `y` first by taking the `ln` of both sides:
`lna^y=lnb^x`
use the fact that:
`logx^y=ylogx`
so that you get:
`yln(a)=xln(b)`
or rearranging:
`y=ln(b)/ln(a)x`Then
we can use the change of base in reverse to get:
`log_a(b)=ln(b)/ln(a)`
and we get:
`y=log_a(b)x`
which is in the form `y=mx`
with `m=log_a(b)="constant"="Slope"`