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 the base \(\sqrt{2}\) log into base 2 before combining terms.

Answer 1

The Correct Option is A

Step 1: Convert both logs to the same base.
The equation is \( \log_2 x + \log_{\sqrt{2}} x = 48 \).
Using the change of base rule, \( \log_{\sqrt{2}} x = \dfrac{\log_2 x}{\log_2 \sqrt{2}} \), and since \( \log_2 \sqrt{2} = \dfrac{1}{2} \), this becomes \( 2 \log_2 x \).

Step 2: Combine and solve for x.
The equation now reads \( \log_2 x + 2\log_2 x = 48 \), so \( 3 \log_2 x = 48 \), giving \( \log_2 x = 16 \).
Rewriting in exponential form gives \( x = 2^{16} \).

Final Answer:
The value of x that satisfies the equation is \( 2^{16} \).\[ \boxed{x = 2^{16}} \]