Question

If log2[3 + log3{4 + log4(x - 1)}] - 2 = 0 then 4x equals

Answer 1

Correct Answer: 5

We are given:

\(\log_2 \left[ 3 + \log_3 \left( 4 + \log_4 (x - 1) \right) \right] - 2 = 0\)

Step 1: Bring 2 to RHS:

\(\log_2 \left[ 3 + \log_3 \left( 4 + \log_4 (x - 1) \right) \right] = 2\)

Step 2: Convert logarithmic to exponential form:

\(3 + \log_3 \left( 4 + \log_4 (x - 1) \right) = 2^2 = 4\)

Step 3: Subtract 3:

\(\log_3 \left( 4 + \log_4 (x - 1) \right) = 1\)

Step 4: Convert logarithmic to exponential again:

\(4 + \log_4 (x - 1) = 3^1 = 3\)

Step 5: Subtract 4:

\(\log_4 (x - 1) = -1\)

Step 6: Convert to exponential:

\(x - 1 = 4^{-1} = \frac{1}{4}\)

Step 7: Solve for x:

\(x = \frac{1}{4} + 1 = \frac{5}{4}\)

Step 8: Find \(4x\):

\(4x = 4 \times \frac{5}{4} = \boxed{5}\)