Proof

Prove if $x,y,$ and $b$ are all positive real values with $b,y \neq 1$, then

$$\frac{\log_b x}{\log_b y} = \log_y x$$



First note that we can evaluate $\log_b x, \log_b y$ and $\log_y x$, since $x,y,$ and $b$ are all positive real values with $b,y \neq 1$. Let us denote the last two values by $m$ and $n$, so that

$$\begin{array}{rcl}
\log_b y &=& m\\
\log_y x &=& n\\
\end{array}$$

If we express the above in exponential form, we have

$$\begin{array}{rcl}
b^m &=& y\\
y^n &=& x\\
\end{array}$$

Note, that we know from the laws of exponents that

$$b^{mn} = (b^m)^n = y^n = x$$

Rewriting the resulting $b^{mn} = x$ in logarithmic form, we have

$$\log_b x = mn$$

Now, we can substitute for $m$ and $n$ to obtain

$$\log_b x = (\log_b y)(\log_y x)$$

Finally, dividing both sides by $\log_b y$ completes the proof.

$$\frac{\log_b x}{\log_b y} = \log_y x$$