Question

\(\log_2(9 - 2^x) = 10^{\log(3-x)}\) solve for \(x\)

• A. 0
• B. 3
• C. both (a) and (b)
• D. 0 and 6

Hint:

Always check domain restrictions for logarithmic functions.

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ 10^{\log(3-x)} = 3-x \]
Step 2: Calculation / Simplification}
\(\log_2(9 - 2^x) = 3 - x\)
\(9 - 2^x = 2^{3-x} = \frac{8}{2^x}\)
Let \(y = 2^x\): \(9 - y = \frac{8}{y} \Rightarrow y^2 - 9y + 8 = 0\)
\((y-1)(y-8) = 0 \Rightarrow y = 1, 8\)
\(2^x = 1 \Rightarrow x = 0\)
\(2^x = 8 \Rightarrow x = 3\)
Check \(x=3\): \(\log_2(9-8) = \log_2 1 = 0\), RHS = \(10^{\log 0}\) (undefined)
\(\therefore x = 0\) only.
Step 3: Final Answer
\[ 0 \]