Question

If $\log_2 x + \log_4 x = 3$, find $x$.

• A. 4
• B. 8
• C. 16
• D. 32

Hint:

Convert logarithms to the same base and solve. Verify by substituting into the original equation.

Answer 1

The Correct Option is A

- Step 1: Rewrite $\log_4 x$. Since $4 = 2^2$, $\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2}$.
- Step 2: Set up equation. Given: $\log_2 x + \frac{\log_2 x}{2} = 3$.
- Step 3: Simplify. Let $y = \log_2 x$. Then:
\[ y + \frac{y}{2} = 3 \implies \frac{2y + y}{2} = 3 \implies \frac{3y}{2} = 3 \implies y = 2. \] - Step 4: Solve for $x$. $\log_2 x = 2 \implies x = 2^2 = 4$.
- Step 5: Verify. For $x = 4$: $\log_2 4 = 2$, $\log_4 4 = 1$, sum = $2 + 1 = 3$. Matches.
- Step 6: Alternative. Use base 4: $\log_2 x = \log_4 (x^2)$. Equation: $\log_4 (x^2) + \log_4 x = \log_4 (x^3) = 3 \implies x^3 = 4^3 = 64 \implies x = 4$.
- Step 7: Check options. $x = 4$ is option (1), - Step 8: Conclusion. Option (1) 4 is correct.