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

1 Answer
Dec 18, 2015

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)xThen
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"