Question

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

Answer 1

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 $4x$: $\ 4x = 4 \times \frac{5}{4} = 5$