Subtracting Logs with the Same Base

Here is a proof that subtracting logs with the same base is equal to taking the same log with the same base of the two numbers divided by each other. Note $\log_a$ means log base $a$. $$\log_a x - \log_a y = \log_a \frac xy.$$

Proof:

Step 1: Let $m = \log_a x$ and $n = \log_a y$.

Step 2: Write in exponent form: $x = a^m$ and $y = a^n$

Step 3: Divide $x$ by $y$: $\frac x y = \frac{a^m}{a^n}= a^{m - n}$.

Step 4: Take $\log_a$ of both sides and evaluate:

$\log_a \frac xy = \log_a a^{m - n}$

$\log_a \frac xy = (m - n) \log_a a$

$\log_a \frac xy = m - n$

$\log_a \frac xy = \log_a x - \log_a y$