Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is D

The expression to evaluate is \(log_5 25 + log_2 (log_3 81)\). 

Let's break it down into steps:

1. First, evaluate \(log_5 25\):
   Since \(25 = 5^2\), we have \(log_5 25 = log_5 (5^2)\).
   Using the property of logarithms \(log_b (a^n) = n \cdot log_b a\), it follows that \(log_5 (5^2) = 2 \cdot log_5 5\).
   Since \(log_5 5 = 1\), we have \(2 \cdot 1 = 2\).
   Thus, \(log_5 25 = 2\).
2. Next, evaluate \(log_3 81\):
   Since \(81 = 3^4\), we have \(log_3 81 = log_3 (3^4)\).
   Using the property of logarithms again, \(log_3 (3^4) = 4 \cdot log_3 3\).
   Since \(log_3 3 = 1\), we have \(4 \cdot 1 = 4\).
   Thus, \(log_3 81 = 4\).
3. Now evaluate \(log_2 (log_3 81)\):
   We found that \(log_3 81 = 4\), so we need \(log_2 4\).
   Since \(4 = 2^2\), \(log_2 4 = log_2 (2^2)\).
   Using the property of logarithms, \(log_2 (2^2) = 2 \cdot log_2 2\).
   Since \(log_2 2 = 1\), we have \(2 \cdot 1 = 2\).
   Thus, \(log_2 4 = 2\).
4. Add the results:
   \(log_5 25 + log_2 (log_3 81) = 2 + 2 = 4\).

The final result is 4.