Question

Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms.

Log base 5 (x^4y^5/2)

Answer 1

4Log₅ x + 5Log₅ y – Log₅ 2

Explanation:

Log₅ (x⁴y⁵/2)

The above expression can be simplified as follow:

Log₅ (x⁴y⁵/2)

Recall:

Log(M/N) = Log M – Log N

Therefore,

Log₅ (x⁴y⁵/2) = Log₅ x⁴y⁵ – Log₅ 2

Recall

Log MN = Log M + Log N

Therefore,

Log₅ x⁴y⁵ – Log₅ 2

= Log₅ x⁴ + Log₅ y⁵ – Log₅ 2

Recall

Log Mⁿ = nLog M

Therefore,

Log₅ x⁴ + Log₅ y⁵ – Log₅ 2

= 4Log₅ x + 5Log₅ y – Log₅ 2

Thus,

Log₅ (x⁴y⁵/2)

= 4Log₅ x + 5Log₅ y – Log₅ 2