Question

The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution

• A. $\dfrac{7}{8}$
• B. $\dfrac{8}{7}$
• C. $\dfrac{-3}{4}$
• D. None of these

Hint:

When bases are equal, equate exponents directly. Always verify the solution against the given domain condition (here $x>0$).

Answer 1

The Correct Option is D

Step 1: Conceptual Understanding:
When the bases on both sides of an exponential equation can be expressed as a power of the same base, we equate the exponents.
Step 2: Explanation in Detail:
Rewrite the right-hand side: $9^{2x-2} = (3^2)^{2x-2} = 3^{4x-4}$.
So the equation becomes $3^{3x+4} = 3^{4x-4}$.
Equating exponents: $3x + 4 = 4x - 4 \Rightarrow x = 8$.
Since $x=8>0$, it is valid. However, $x=8$ does not match any of the given options (A), (B), or (C).
Step 3: Therefore, Stating the Final Answer
The solution is $x = 8$, which corresponds to option (D) None of these.