Question

If  \(log_2[3 + log_ 3[4 + log_4(x - 1)] - 2 = 0\) then 4x equals

Answer 1

We are given the logarithmic equation:

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

Step 1: Rearranging and simplifying the equation

Rearranging the equation, we get: \[ \log_2 \left[ 3 + \log_3 \left( 4 + \log_4 (x - 1) \right) \right] = 2 \]

Step 2: Apply logarithmic properties

Using the properties of logarithms, we get: \[ 3 + \log_3 \left( 4 + \log_4 (x - 1) \right) = 2^2 \] Simplifying: \[ 3 + \log_3 \left( 4 + \log_4 (x - 1) \right) = 4 \]

Step 3: Isolate the logarithmic term

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

Step 4: Solve the inner logarithmic equation

This implies: \[ 4 + \log_4 (x - 1) = 3 \] Subtracting 4 from both sides: \[ \log_4 (x - 1) = -1 \]

Step 5: Apply logarithmic properties

Using the properties of logarithms, we get: \[ x - 1 = 4^{-1} \] Simplifying: \[ x - 1 = \frac{1}{4} \]

Step 6: Solve for \( x \)

Adding 1 to both sides: \[ x = \frac{5}{4} \]

Step 7: Find \( 4x \)

To find \( 4x \), we multiply: \[ 4x = 4 \times \frac{5}{4} = 5 \]

Final Answer:

The value of \( 4x \) is \( \boxed{5} \).