Question

If \( \log_{25}[5 \log_3 (1 + \log_3(1 + 2 \log_2 x))] = 1/2 \) then x is

• A. 4
• B. 16
• C. 8
• D. 2

Hint:

When solving nested logarithmic equations, think of it like peeling an onion. Start with the outermost function and apply the inverse operation (in this case, exponentiation) to remove one layer at a time until you isolate the variable.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves solving a nested logarithmic equation. We need to work from the outermost logarithm inwards, using the fundamental definition of a logarithm at each step.
Step 2: Key Formula or Approach:
The definition of a logarithm states that if \( \log_b(y) = z \), then \( y = b^z \). We will apply this rule repeatedly.
Step 3: Detailed Explanation:
The given equation is \( \log_{25}[5 \log_3 (1 + \log_3(1 + 2 \log_2 x))] = 1/2 \). Step 1: Outermost Logarithm (base 25) Using the definition \( \log_b(y) = z \implies y = b^z \): \[ 5 \log_3 (1 + \log_3(1 + 2 \log_2 x)) = 25^{1/2} \] Since \( 25^{1/2} = \sqrt{25} = 5 \): \[ 5 \log_3 (1 + \log_3(1 + 2 \log_2 x)) = 5 \] Divide both sides by 5: \[ \log_3 (1 + \log_3(1 + 2 \log_2 x)) = 1 \] Step 2: Second Logarithm (base 3) Applying the definition again: \[ 1 + \log_3(1 + 2 \log_2 x) = 3^1 \] \[ 1 + \log_3(1 + 2 \log_2 x) = 3 \] Subtract 1 from both sides: \[ \log_3(1 + 2 \log_2 x) = 2 \] Step 3: Third Logarithm (base 3) Applying the definition again: \[ 1 + 2 \log_2 x = 3^2 \] \[ 1 + 2 \log_2 x = 9 \] Subtract 1 from both sides: \[ 2 \log_2 x = 8 \] Divide by 2: \[ \log_2 x = 4 \] Step 4: Innermost Logarithm (base 2) Applying the definition one last time: \[ x = 2^4 \] \[ x = 16 \] Step 4: Final Answer:
The value of x is 16.