Question:medium
If \(\log_{a^{4}}65536=2\), what is the value of 'a'?
If \(\log_{a^{4}}65536=2\), what is the value of 'a'?
Updated On: Nov 25, 2025
- 2
- 4
- 6
- 8
- √2
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The Correct Option is B
Solution and Explanation
The correct answer is option (B):
4
Let's break down this logarithmic equation step-by-step to find the value of 'a'. The equation is log_{a^4} 65536 = 2.
Remember the fundamental relationship between logarithms and exponents: log_b x = y is equivalent to b^y = x.
Applying this to our problem, we can rewrite the logarithmic equation as an exponential equation: (a^4)^2 = 65536.
Now, simplify the left side: a^(4*2) = 65536, which simplifies to a^8 = 65536.
We need to find the eighth root of 65536 to isolate 'a'. Recognizing that 65536 is a power of 2, let's express it that way. 65536 = 2^16.
So our equation now is: a^8 = 2^16.
To solve for 'a', take the eighth root of both sides, or rewrite the right side so it looks like a power of 8. We can rewrite 2^16 as (2^2)^8 since 2*8 = 16. This gives us a^8 = (2^2)^8, which is a^8 = 4^8.
Therefore, a = 4. The value of 'a' that satisfies the original equation is 4.
Remember the fundamental relationship between logarithms and exponents: log_b x = y is equivalent to b^y = x.
Applying this to our problem, we can rewrite the logarithmic equation as an exponential equation: (a^4)^2 = 65536.
Now, simplify the left side: a^(4*2) = 65536, which simplifies to a^8 = 65536.
We need to find the eighth root of 65536 to isolate 'a'. Recognizing that 65536 is a power of 2, let's express it that way. 65536 = 2^16.
So our equation now is: a^8 = 2^16.
To solve for 'a', take the eighth root of both sides, or rewrite the right side so it looks like a power of 8. We can rewrite 2^16 as (2^2)^8 since 2*8 = 16. This gives us a^8 = (2^2)^8, which is a^8 = 4^8.
Therefore, a = 4. The value of 'a' that satisfies the original equation is 4.
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