Question

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

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

Convert the log equation to exponent form and write 65536 as a power of 2.

Answer 1

The Correct Option is B

Step 1: Rewrite the log equation in exponent form.
\( \log_{a^4} 65536 = 2 \) means \( (a^4)^2 = 65536 \).

Step 2: Simplify the left side.
\( (a^4)^2 = a^8 \), so \( a^8 = 65536 \).

Step 3: Express 65536 as a power of 2.
\( 65536 = 2^{16} \), since \( 2^{16} = 65536 \).

Step 4: Equate the exponents.
\( a^8 = 2^{16} \), so \( a = 2^{16/8} = 2^2 = 4 \).

Final Answer:
The value of a is 4. \[ \boxed{a = 4} \]