Question

solve. simplify your answer(s). if there are multiple answers, separate them with commas.\\(\log_{7}(7w - 12) = \log_{7}4w\\)\\(w = \square\\)

Explanation:

Step1: Use the property of logarithms

If $\log_b a = \log_b c$, then $a = c$ (for $b>0, b
eq1, a>0, c>0$). So we can set the arguments equal:
$7w - 12 = 4w$

Step2: Solve for \( w \)

Subtract \( 4w \) from both sides:
$7w - 4w - 12 = 0$
$3w - 12 = 0$

Add 12 to both sides:
$3w = 12$

Divide both sides by 3:
$w = \frac{12}{3} = 4$

We should also check the domain. For $\log_7(7w - 12)$ and $\log_7(4w)$, we need $7w - 12>0$ and $4w>0$.
For $w = 4$: $7(4)-12 = 28 - 12 = 16>0$ and $4(4)=16>0$, so it is valid.

Answer:

4