Question

If a positive real \(x\) satisfies the following equation
\[ \log_2 x + \log_{\sqrt{2}} x = 48, \]
then the value of \(x\) is _________________

• A. \(2^{16}\)
• B. \(4^{16}\)
• C. \(2^{14}\)
• D. \(4^{14}\)

Hint:

Convert log base sqrt(2) to base 2 using the change-of-base rule before combining terms.

Answer 1

The Correct Option is A

Step 1: Set up using natural logarithms.
Let $x$ be the unknown positive number. Write both logarithms using natural log: $\log_2 x = \dfrac{\ln x}{\ln 2}$ and $\log_{\sqrt{2}} x = \dfrac{\ln x}{\ln \sqrt{2}}$.

Step 2: Simplify $\ln \sqrt{2}$.
Since $\sqrt{2} = 2^{1/2}$, we get $\ln \sqrt{2} = \dfrac{1}{2}\ln 2$. So
\[ \log_{\sqrt{2}} x = \frac{\ln x}{\frac{1}{2}\ln 2} = \frac{2\ln x}{\ln 2} \]
Step 3: Add the two terms.
\[ \frac{\ln x}{\ln 2} + \frac{2\ln x}{\ln 2} = 48 \] \[ \frac{3\ln x}{\ln 2} = 48 \] \[ \ln x = 16 \ln 2 \]
Step 4: Solve for $x$.
\[ \ln x = \ln(2^{16}) \] so \[ x = 2^{16} \] This matches the value obtained from the base-2 approach, confirming the answer.

Final Answer:
\[ x = 2^{16} \]