Question

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solve the equation. \\(\log_{7}(x + 4) - \log_{7}x = 2\\) select the correct choice below and fill in any answer boxes present in your choice. a. \\(x = \square\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. use a comma to separate answe 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:
$\log_7 \frac{x + 4}{x} = 2$

Step2: Convert to exponential form

Recall that $\log_b y = k$ is equivalent to $b^k = y$. So, $7^2 = \frac{x + 4}{x}$

Step3: Solve the resulting equation

Calculate $7^2 = 49$, so $49 = \frac{x + 4}{x}$. Multiply both sides by $x$ (assuming $x
eq 0$):
$49x = x + 4$
Subtract $x$ from both sides:
$48x = 4$
Divide both sides by 48:
$x = \frac{4}{48} = \frac{1}{12}$

Step4: Check domain

For $\log_7 (x + 4)$ and $\log_7 x$ to be defined, $x + 4 > 0$ and $x > 0$. Substituting $x = \frac{1}{12}$, we have $\frac{1}{12} + 4 > 0$ and $\frac{1}{12} > 0$, so it's valid.

Answer:

$\frac{1}{12}$