Select the correct answer.Which expression is equivalent to 5⁢log𝑥⁡2−log𝑥⁡4? A. log𝑥⁡6 B. log𝑥⁡(52) C. log𝑥⁡28 D. log𝑥⁡8

Sure, let's solve the expression step by step.

We need to simplify the expression $5 \log_x 2 - \log_x 4$.

Step 1: Use the power rule of logarithms. The power rule states that $a \log_b c = \log_b (c^a)$. Applying this rule to the first term:

$5 \log_x 2 = \log_x (2^5) = \log_x 32$

So the expression now is:

$\log_x 32 - \log_x 4$

Step 2: Use the quotient rule of logarithms. The quotient rule states that $\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)$. Applying this rule to the expression:

$\log_x 32 - \log_x 4 = \log_x \left( \frac{32}{4} \right)$

Step 3: Simplify the fraction inside the logarithm.

$\frac{32}{4} = 8$

So the expression simplifies to:

$\log_x 8$

Therefore, the correct answer is:

D. $\log_x 8$