1/2(log5^x+log5^y)-2log5(x+1)

Question

write the logarithmic expression as a single expression condense

Answer 1

log(5^x)=xlog(5); log(5^y)=ylog(5).

The first two terms can be combined: ½(x+y)log(5)=log√5^x+y.

"log5(x+1)" could mean

(1) log(5(x+1)), 2log(5(x+1))=log(5²(x+1)²)=log(25(x+1)²); or

(2) 2log(5)(x+1)=(x+1)log(25)=log(25^x+1)=log(5^2(x+1))=log(5^2x+2)

Therefore the whole expression becomes:

(1) log(√5^x+y/(25(x+1)²)), log(5^½x+½y/(5²(x+1)²)), log(5^½x+½y-2/(x+1)²;

(2) log(√5^x+y/5^2x+2), log(5^½x+½y-2x-2), log(5^½y-(3/2)x-2), log√(5^y-3x-4).