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

To solve the expression \(\log_5 25 + \log_2(\log_3 81)\), we need to evaluate each term separately and then add the results.

Evaluate \(\log_5 25\): 

The logarithm \(\log_b a\) is defined as the power to which the base \(b\) must be raised to give the number \(a\). Therefore, we can rewrite:

\(\log_5 25 = x \Rightarrow 5^x = 25\)

Since \(25 = 5^2\), we have:

\(\log_5 25 = 2\)

Evaluate \(\log_3 81\):

Similarly, rewrite the term as:

\(\log_3 81 = y \Rightarrow 3^y = 81\)

Since \(81 = 3^4\), we have:

\(\log_3 81 = 4\)

Evaluate \(\log_2(\log_3 81)\):

Since \(\log_3 81 = 4\), we need to find \(\log_2 4\):

\(\log_2 4 = z \Rightarrow 2^z = 4\)

And since \(4 = 2^2\), we have:

\(\log_2 4 = 2\)

Add both results:

\(\log_5 25 + \log_2(\log_3 81) = 2 + 2 = 4\)

Thus, the value of \(\log_5 25 + \log_2(\log_3 81)\) is 4.

Therefore, the correct answer is: \(4\).

Answer 2

Step 1: Simplify the inner logarithm first.
\(\log_3 81\) asks "3 raised to what power gives 81". Since \(81 = 3^4\), \(\log_3 81 = 4\).

Step 2: Simplify the outer logarithm using this result.
The expression becomes \(\log_5 25 + \log_2 4\). For \(\log_5 25\), since \(25 = 5^2\), \(\log_5 25 = 2\). For \(\log_2 4\), since \(4 = 2^2\), \(\log_2 4 = 2\).

Step 3: Add the two results.
\(\log_5 25 + \log_2(\log_3 81) = 2 + 2 = 4\).

Final Answer:
The value of the expression is 4. \[ \boxed{4} \]