Which of the following expressions is equal to log
(x sqrt-y)/z^5
A. log x + log (1/2) + log y– log 5 – log z
B. log [x + (1/2)y – 5z]
C. log x + (1/2)log y – 5 log z
d. [(1/2) log x log y]/(5 log z)
C, with some ( ) in the numerator is (xy^1/2)/Z^5 .
To determine which expression is equal to "log((x√-y)/z^5)", we need to understand the properties of logarithms.
1. The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator:
log(a/b) = log(a) - log(b)
2. The logarithm of a power is equal to the product of the logarithm and the exponent:
log(a^n) = n * log(a)
Now let's analyze each option:
A. log x + log (1/2) + log y – log 5 – log z
This expression doesn't match the given expression since it doesn't have the square root (sqrt) term and doesn't involve z^5 as the denominator.
B. log [x + (1/2)y – 5z]
This expression doesn't match the given expression since it has an addition in the argument of the logarithm instead of a quotient.
C. log x + (1/2)log y – 5 log z
This expression matches the given expression, as it follows the property of logarithms where the square root term is expressed as (1/2)log y instead of log √y, and z^5 is correctly expressed as 5 log z.
D. [(1/2) log x log y]/(5 log z)
This expression doesn't match the given expression since it involves the product of logarithms in the numerator, which is not equivalent to the given expression.
Therefore, the correct option is C. log x + (1/2)log y – 5 log z.