Question

\(\log_5 25 + \log_2(\log_3 81)\) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Simplify log 3 of 81 first, then use that value inside log base 2, and add log 5 of 25.

Answer 1

The Correct Option is D

Step 1: Use the rule $\log_a(a^n) = n$ directly.
Write each number as a power of its own base. $25 = 5^2$, so $\log_5 25 = 2$ right away by this rule, no separate power step needed. Similarly $81 = 3^4$, so $\log_3 81 = 4$.

Step 2: Substitute the inner value into the outer log.
The expression is now $2 + \log_2 4$. Since $4 = 2^2$, the same rule gives $\log_2 4 = 2$.

Step 3: Verify using natural log as a cross check.
$\log_5 25 = \frac{\ln 25}{\ln 5} = \frac{2\ln 5}{\ln 5} = 2$. $\log_3 81 = \frac{\ln 81}{\ln 3} = \frac{4\ln 3}{\ln 3} = 4$. $\log_2 4 = \frac{\ln 4}{\ln 2} = \frac{2\ln 2}{\ln 2} = 2$. All match the quicker method.

Step 4: Add the final numbers.
$2 + 2 = 4$.

Final Answer:
The expression equals 4. \[ \boxed{4} \]