Find the value of log64(l/16).a)-2/3b)2/3c)3/2d)-3/2Correct answer is ...
To find the value of log64(l/16), we need to simplify the expression first and then evaluate the logarithm.
1. Simplifying the expression:
- We can rewrite l/16 as l * (1/16).
- Since log(a * b) = log(a) + log(b), we can rewrite log64(l/16) as log64(l) + log64(1/16).
2. Evaluating the logarithm:
- We know that loga(b) = c can be rewritten as a^c = b.
- Using this property, we can rewrite log64(l) + log64(1/16) as 64^c = l and 64^d = 1/16 respectively.
- We need to find the values of c and d that satisfy these equations.
3. Simplifying the equations:
- 64^c = l can be rewritten as 2^6c = l.
- 64^d = 1/16 can be rewritten as 2^6d = 1/2^4.
4. Solving for c and d:
- From the equation 2^6c = l, we can equate the exponents: 6c = 1.
Solving for c, we find c = 1/6.
- From the equation 2^6d = 1/2^4, we can equate the exponents: 6d = -4.
Solving for d, we find d = -4/6 = -2/3.
5. Substituting the values of c and d into the expression:
- log64(l) + log64(1/16) = log64(2^6c) + log64(2^6d)
= 6c + 6d
= 6 * (1/6) + 6 * (-2/3)
= 1 - 4
= -3.
Therefore, the value of log64(l/16) is -3. However, none of the given options match this result. So, there might be an error in the options provided.