Question

If \( \log_{10} 2,\ \log_{10}(2^{x} - 1),\ \log_{10}(2^{x} + 3) \) are in A.P., then \( x \) is:

• A. $\log_{2} 5$
• B. $\log_{5} 2$
• C. $\log_{10} 5$
• D. $\log_{2} 10$

Hint:

Logarithmic properties $\log a + \log c = \log(ac)$ and $n \log a = \log(a^{n})$ are essential here.

Answer 1

The Correct Option is A

Step 1: Concept
If $a, b, c$ are in A.P., then $2b = a + c$.
Step 2: Analysis
$2 \log(2^{x}-1) = \log 2 + \log(2^{x}+3) = \log[2(2^{x}+3)]$.
$(2^{x}-1)^{2} = 2(2^{x}+3)$. Let $2^{x} = y$.
$(y-1)^{2} = 2(y+3) \Rightarrow y^{2} - 2y + 1 = 2y + 6 \Rightarrow y^{2} - 4y - 5 = 0$.
$(y-5)(y+1) = 0 \Rightarrow y = 5$ ($y = -1$ rejected as $2^{x}>0$).
Step 3: Conclusion
$2^{x} = 5 \Rightarrow x = \log_{2} 5$.
Final Answer: (A)