Question:easy

If \( \log_{a^4} 65536 = 2 \), what is the value of 'a'?

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Convert the log equation to exponent form and write 65536 as a power of 2.
Updated On: Jul 21, 2026
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The Correct Option is B

Solution and Explanation

Step 1: Apply the change of base idea directly to the log definition.
$ \log_{a^4} 65536 = 2 $ means $ \dfrac{\log 65536}{\log a^4} = 2 $.

Step 2: Use the power rule on the denominator.
$ \log a^4 = 4 \log a $, so the equation becomes $ \log 65536 = 8 \log a $.

Step 3: Break 65536 into a power of 2.
$ 65536 = 2^{16} $, so $ \log 65536 = 16 \log 2 $.

Step 4: Solve for log a.
$ 8 \log a = 16 \log 2 $, so $ \log a = 2 \log 2 = \log 4 $.

Step 5: Remove the log.
Since $ \log a = \log 4 $, it follows that $ a = 4 $.

Final Answer:
The value of a is 4. $ a = 4 $
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