Question

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

Answer 1

Correct Answer: 5

Given:
\(\log_2 \left[ 3 + \log_3 \left( 4 + \log_4 (x-1) \right) \right] - 2 = 0\)
Now, rearranging and simplifying: 
\(\log_2 \left[ 3 + \log_3 \left( 4 + \log_4 (x-1) \right) \right] = 2\)

Using the properties of logarithm: \(3 + \log_3 \left( 4 + \log_4 (x-1) \right) = 2^2\)
\(3 + \log_3 \left( 4 + \log_4 (x-1) \right) = 4\)

Subtracting 3 from both sides:
\(\log_3 \left( 4 + \log_4 (x-1) \right) = 1\)

This implies: \(4 + \log_4 (x-1) = 3\) 
\(\log_4 (x-1) = -1\)

Now, using the properties of logarithm:
\(x-1 = 4^{-1}\)

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

Now, adding 1 to both sides: 
\(x = \frac{5}{4}\)

To find \(4x\): \(\ 4x = 4 \times \frac{5}{4} = 5\)