Question

Are the following equivalent expressions? Explain why or why not.

2 log₅x - (4 log₅y + 3 log₅2) = log₅ (x² / 6y⁴)
A) Yes, they are equivalent because of the properties of logarithms.
B) No, they are not equivalent due to different bases.
C) Yes, they are equivalent for certain values of x and y.
D) No, they are not equivalent for any values of x and y.

Answer 1

The expressions are not equivalent. After applying the properties of logarithms, the denominators differ (8y⁴ vs. 6y⁴), which makes the expressions unequal.

Step-by-step explanation:

The question asks if the following expressions are equivalent:

2 log₅x - (4 log₅y + 3 log₅2) = log₅ (x² / 6y⁴)

By applying the properties of logarithms, we can simplify the expression on the left side:

Use the property that the logarithm of a power is the exponent times the logarithm of the base, which gives us '2 log₅x' = 'log₅ x² and '4 log₅y' = 'log₅ y⁴'

However, the expression on the right side is 'log₅ (x² / 6y⁴)'. The numerators match, but the denominators do not, as 8y⁴ is not the same as 6y⁴.

Therefore, the expressions are not equivalent because their denominators differ, meaning option B is the correct answer: No, they are not equivalent due to different denominators. The bases of the logarithms are not an issue, since they are consistent across the expressions.