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

Question

If the domain of \[ f(x)=\log_{(10x^2-17x+7)}\,(18x^2-11x+1) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then find $90(a+b+c+d+e)$.

Answer 1

Given:

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

Step 1: Isolate the main logarithm:

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

Step 2: Convert to exponential form (base 2):

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

Step 3: Subtract 3:

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

Step 4: Convert to exponential form (base 3):

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

Step 5: Subtract 4:

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

Step 6: Convert to exponential form (base 4):

$x - 1 = 4^{-1} = \frac{1}{4}$

Step 7: Solve for x:

$x = \frac{1}{4} + 1 = \frac{5}{4}$

Step 8: Calculate $4x$:

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

Correct Answer: 5