Question

solve the equation.
\\(\log_4(x + 9) - \log_4 x = 2\\)
select the correct choice below and fill in any answer
a. \\(x = \\)
(simplify your answer, including any radicals.
b. there is no solution.

Explanation:

Step1: Apply log subtraction rule

Using the property $\log_b M - \log_b N = \log_b \frac{M}{N}$, we rewrite the equation:
$\log_4 \frac{x + 9}{x} = 2$

Step2: Convert to exponential form

Recall that $\log_b y = k$ implies $b^k = y$. So, $4^2 = \frac{x + 9}{x}$

Step3: Simplify and solve for x

$16 = \frac{x + 9}{x}$
Multiply both sides by $x$ (assuming $x
eq 0$):
$16x = x + 9$
Subtract $x$ from both sides:
$15x = 9$
Divide by 15:
$x = \frac{9}{15} = \frac{3}{5}$

Step4: Check domain

For $\log_4 (x + 9)$ and $\log_4 x$ to be defined, $x + 9 > 0$ and $x > 0$.
$x = \frac{3}{5} > 0$ and $\frac{3}{5} + 9 = \frac{48}{5} > 0$, so it's valid.

Answer:

A. $x = \frac{3}{5}$