Question

Use the properties of logarithms to expand the expression as a sum or difference, and/or constant multiple of logarithms. (Assume all variables are positive) ​

Answer 1

`\log_5(9)+(1)/(2)\log_5(x)-3\log_5(y)`

Explanation:

`\log_5((9√(x))/(y^3))`

First rule I'm going to use is the quotient rule:

`\log_b((m)/(n))=\log_b(m)-\log_b(n)`

`\log_5(9√(x))-\log_5(y^3)`

Secondly, I'm going to rewrite the radical.

`√(x)=x^(1)/(2)`

`\log_5(9x^(1)/(2))-\log_5(y^3)`

Third, I'm going to use the product rule on the first term:

`\log_b(mn)=\log_b(m)+\log_b(n)`

`\log_5(9)+\log_5(x^(1)/(2))-\log_5(y^3)`

Fourth, I'm going to use power rule for both of the last two terms:

`\log_b(m^r)=r\log_b(m)`

`\log_5(9)+(1)/(2)\log_5(x)-3\log_5(y)`