Examkin

Question

Difficulty: Very hardLogarithms and Change of Base

If log⁡x2⋅log⁡x162=log⁡x642\log_x 2 \cdot \log_{\frac{x}{16}} 2 = \log_{\frac{x}{64}} 2, what is the sum of all possible real values of xx?

  1. A
    5
  2. 12Answer
  3. C
    32
  4. D
    66

Answer

The sum of all possible real values of xx is 12.
Applying the reciprocal base formula log⁡ab=1log⁡ba\log_a b = \frac{1}{\log_b a} allows us to express the equation as 1log⁡2x⋅1log⁡2x−4=1log⁡2x−6\frac{1}{\log_2 x} \cdot \frac{1}{\log_2 x - 4} = \frac{1}{\log_2 x - 6}. Letting u=log⁡2xu = \log_2 x, we obtain 1u(u−4)=1u−6\frac{1}{u(u-4)} = \frac{1}{u-6}, which simplifies to the quadratic equation u2−5u+6=0u^2 - 5u + 6 = 0. Factoring gives u=2u = 2 or u=3u = 3. Solving for x=2ux = 2^u yields x=22=4x = 2^2 = 4 and x=23=8x = 2^3 = 8. The sum of these values is 4+8=124 + 8 = 12.

Step-by-Step Solution

1
Apply the change of base rule log⁡ab=1log⁡ba\log_a b = \frac{1}{\log_b a} to convert all logarithms to base 2.
\frac{1}{\log_2 x} \cdot \frac{1}{\log_2\left(\frac{x}{16}\right)} = \frac{1}{\log_2\left(\frac{x}{64}\right)}
Converting terms to a common base (base 2) simplifies the variable terms in the denominators.
2
Use the quotient rule of logarithms log⁡b(MN)=log⁡bM−log⁡bN\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N and substitute u=log⁡2xu = \log_2 x.
\log_2\left(\frac{x}{16}\right) = u - 4 \quad \text{and} \quad \log_2\left(\frac{x}{64}\right) = u - 6
Since log⁡216=4\log_2 16 = 4 and log⁡264=6\log_2 64 = 6, substituting u=log⁡2xu = \log_2 x transforms the expression into an algebraic equation in terms of uu.
3
Form and solve the algebraic equation for uu.
\frac{1}{u(u - 4)} = \frac{1}{u - 6} \implies u(u - 4) = u - 6 \implies u^2 - 5u + 6 = 0
Equating the denominators yields a quadratic equation u2−5u+6=0u^2 - 5u + 6 = 0.
4
Factor the quadratic equation to find uu, then solve for xx.
(u - 2)(u - 3) = 0 \implies u = 2 \text{ or } u = 3 \implies x = 2^2 = 4 \text{ or } x = 2^3 = 8
Converting back from u=log⁡2xu = \log_2 x gives the real roots x=4x = 4 and x=8x = 8.
5
Calculate the sum of the valid solutions.
4 + 8 = 12
The question asks specifically for the sum of all possible real values of xx.

Key Concept

Logarithm Change of Base Formula and Logarithmic Equations
Estimated Time:2m 0s
Rate this question

Topics

Logarithms and Change of Base

More questions from this topic