Question

Expand the following logs:


`log_(7) \sqrt{a^(3)b^(9) }`


`log_(6) ((x^(5) )/(y^(9) ) )`


`log_(8) (x^(3) y^(7) )`

Answer 1

QUESTION 1

The given logarithm is

`\log_7√(a^3b^9)`

We rewrite to obtain;

`\log_7(a^3b^9)^{(1)/(2)}`

Use the power rule of logarithm ;
`\log_a(m^n)=n\log_a(m)`

`(1)/(2)\log_7(a^3b^9)`

Use the product rule;
`\log_a(mn)=\log_a(m)+\log_a(n)`

`(1)/(2)[\log_7(a^3)+\log_7(b^9)]`

Use the power rule of logarithms again;

`(1)/(2)[3\log_7(a)+9\log_7(b)]`

Or

`(3)/(2)\log_7(a)+(9)/(2)\log_7(b)]`

QUESTION 2

Given;

`\log_6((x^5)/(y^9))`

Apply the quotient rule of logarithm;
`\log_a(m)-\log_a(n)=\log_a((m)/(n) )`

`\log_6((x^5)/(y^9))=\log_6(x^5)-\log_6(y^9)`

Apply the power rule to get;

`\log_6((x^5)/(y^9))=5\log_6(x)-9\log_6(y)`

QUESTION 3

Given;

`\log_8(x^3y^7)`

Use the product rule to get;

`=\log_8(x^3)+\log_8(y^7)`

Use the power rule now;

`=3\log_8(x)+7\log_8(y)`