Evaluate log5($y^2x^5 ÷ 125b½)$

Analyze the question: We need to expand log₅(y²x⁵ ÷ 125b^½) using the laws of logarithms. The expression is read as y²x⁵ divided by (125 × b^½), which is the interpretation consistent with the options.

Laws needed:

• log(MN) = log M + log N
• log(M/N) = log M − log N
• log(Mⁿ) = n log M
• log₅5 = 1, so log₅125 = log₅5³ = 3

Solution:

log₅( y²x⁵ / (125·b^½) )

= log₅(y²x⁵) − log₅(125·b^½)

= log₅y² + log₅x⁵ − [log₅125 + log₅b^½]

= 2log₅y + 5log₅x − 3 − ½log₅b

Compare with the options:

• A: has "5log₅y²" and no b term, so it is wrong.
• B: has +3 and no b term, so it is wrong.
• C: is not an expansion of this expression, so it is wrong.
• D: 2log₅y + 5log₅x − ½log₅b − 3, which matches our working exactly.

Tip: When a term sits in the denominator, every log from it gets a minus sign. Distribute the negative across everything in the denominator (here both 125 and b^½). Also remember that log₅125 = 3 because 5³ = 125.