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

Dec 18, 2015

I used some properties of logs:

#### Explanation:

Let us try to isolate $y$ first by taking the $\ln$ of both sides:
$\ln {a}^{y} = \ln {b}^{x}$
use the fact that:
$\log {x}^{y} = y \log x$
so that you get:
$y \ln \left(a\right) = x \ln \left(b\right)$
or rearranging:
$y = \ln \frac{b}{\ln} \left(a\right) x$Then
we can use the change of base in reverse to get:
${\log}_{a} \left(b\right) = \ln \frac{b}{\ln} \left(a\right)$
and we get:
$y = {\log}_{a} \left(b\right) x$
which is in the form $y = m x$
with $m = {\log}_{a} \left(b\right) = \text{constant"="Slope}$