Condense the following logs into a single log: 8log_(g) x+5log_(g) y 8log_(5) x+(3)/(4) log_(5) y-5log_(5) z

Question

Condense the following logs into a single log:


`8log_(g) x+5log_(g) y`


`8log_(5) x+(3)/(4) log_(5) y-5log_(5) z`

Answer 1

QUESTION 1

The given logarithm is

`8\log_g(x)+5\log_g(y)`

We apply the power rule of logarithms;
`n\log_a(m)=\log_(m^n)`

`=\log_g(x^8)+\log_g(y^5)`

We now apply the product rule of logarithm;

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

`=\log_g(x^8y^5)`

QUESTION 2

The given logarithm is

`8\log_5(x)+(3)/(4)\log_5(y)-5\log_5(z)`

We apply the power rule of logarithm to get;

`=\log_5(x^8)+\log_5(y^{(3)/(4)})-\log_5(z^5)`

We apply the product to obtain;

`=\log_5(x^8* y^{(3)/(4)})-\log_5(z^5)`

We apply the quotient rule;
`\log_a(m)-\log_a(n)=\log_a((m)/(n) )`

`=\log_5(\frac{x^8* y^{(3)/(4)}}{z^5})`

`=\log_5(\frac{x^8 \sqrt[4]{y^3} }{z^5})`