Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer.
\log_3 (x + 21) - \log_3 (x - 5) = 3
rewrite the given equation without logarithms. do not solve for x.

Explanation:

Step1: Apply log subtraction rule

Using the property $\log_b M - \log_b N = \log_b \frac{M}{N}$, we rewrite the left - hand side of the equation $\log_3(x + 21)-\log_3(x - 5)=3$ as $\log_3\frac{x + 21}{x - 5}=3$.

Step2: Convert log to exponential form

Recall the definition of a logarithm: if $\log_b y = k$, then $y=b^k$. Here, $b = 3$, $y=\frac{x + 21}{x - 5}$, and $k = 3$. So we can convert the logarithmic equation $\log_3\frac{x + 21}{x - 5}=3$ to the exponential equation $\frac{x + 21}{x - 5}=3^3$.

Step3: Simplify the right - hand side

Since $3^3=27$, the equation becomes $\frac{x + 21}{x - 5}=27$.

Answer:

$\frac{x + 21}{x - 5}=27$