Question

Express the given quantity as a single logarithm and simplify: 3logx+2log(y-2)-5logx

Answer 1

Simplify each term.
Simplify 3log(x) by moving 3 inside the logarithm. 
log(x^3)+2log(y−1)−5log(x) 

Simplify 2log(y−1) by moving 2 inside the logarithm. 
log(x^3)+log((y−1)^2)−5log(x) 

Rewrite (y−1)^2 as (y−1)(y−1). 
log(x^3)+log((y−1)(y−1))−5log(x) 

Expand (y−1)(y−1) using the FOIL Method. 
log(x^3)+log(y(y)+y(−1)−1(y)−1(−1))−5log(x) 

Simplify each term. 
log(x^3)+log(y^2−2y+1)+log(x^−5) 

Remove the negative exponent by rewriting x^−5 as 1/x^5. 
log(x^3)+log(y^2−2y+1)+log(1/x^5) 

Combine logs to get log(x^3(y^2−2y+1))
log(x^3(y^2−2y+1))+log(1/x^5)

Combine logs to get log(x^3(y^2−2y+1)/x^5) 
log(x^3(y^2−2y+1)/x^5)

Cancel x^3 in the numerator and denominator. 
log(y^2−2y+1/x^2) 

Rewrite 1 as 1^2. 
y^2−2y+1^2/x^2

Factor by perfect square rule. 
(y−1)^2/x^2

Replace into larger expression. 

log((y−1)^2/x^2)