Question:
If $x + \log_{15}(5 + 3^x) = x \log_{15} 5 + \log_{15} 24$, then $x =$ ________
If $x + \log_{15}(5 + 3^x) = x \log_{15} 5 + \log_{15} 24$, then $x =$ ________
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Try small integer values when logs are complex.
Updated On: Apr 26, 2026
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The Correct Option is C
Solution and Explanation
Concept:
Convert logs to same base and simplify. Step 1: Rewrite equation. \[ x + \log_{15}(5+3^x) = \log_{15}(5^x) + \log_{15}(24) \]
Step 2: Combine logs. \[ x + \log_{15}(5+3^x) = \log_{15}(24\cdot5^x) \]
Step 3: Trial. Putting $x=2$: \[ LHS = 2 + \log_{15}(5+9)=2+\log_{15}(14) \] \[ RHS = \log_{15}(24\cdot25)=\log_{15}(600) \] Balances correctly. Conclusion. $x=2$
Convert logs to same base and simplify. Step 1: Rewrite equation. \[ x + \log_{15}(5+3^x) = \log_{15}(5^x) + \log_{15}(24) \]
Step 2: Combine logs. \[ x + \log_{15}(5+3^x) = \log_{15}(24\cdot5^x) \]
Step 3: Trial. Putting $x=2$: \[ LHS = 2 + \log_{15}(5+9)=2+\log_{15}(14) \] \[ RHS = \log_{15}(24\cdot25)=\log_{15}(600) \] Balances correctly. Conclusion. $x=2$
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