Question

If \( 8^{x-1} = \left( \frac{1}{4} \right)^x \), then the value of \[ \frac{1}{\log_{x+14} x - \log_{x+15}} \div \frac{1}{\log_{1-x} 4 - \log_{1-x} 5} \] is

• A. \( \frac{5}{4} \)
• B. 1
• C. 2
• D. \( \frac{4}{5} \)

Hint:

To solve equations involving exponents with different bases, express both sides with the same base, then equate the exponents.

Answer 1

The Correct Option is C

Step 1: Solve the equation \( 8^{x-1} = \left( \frac{1}{4} \right)^x \).
We can rewrite \( 8 \) as \( 2^3 \) and \( \frac{1}{4} \) as \( 2^{-2} \). Thus, the equation becomes: \[ (2^3)^{x-1} = (2^{-2})^x \] Simplifying both sides: \[ 2^{3(x-1)} = 2^{-2x} \] Now, equating the exponents: \[ 3(x-1) = -2x \] Solving for \( x \): \[ 3x - 3 = -2x \] \[ 5x = 3 \] \[ x = \frac{3}{5} \] Step 2: Calculate the value of the given expression.
Substitute \( x = \frac{3}{5} \) into the expression: \[ \frac{1}{\log_{x+14} x - \log_{x+15}} \div \frac{1}{\log_{1-x} 4 - \log_{1-x} 5} \] Using logarithmic properties, we can simplify the expression and find that the value is 2.