loga xy^3=9; loga x^2y=8

Question

Find value of loga (xy)^(1/2)

Answer 1

loga xy^3+loga x^2y=9+8

Using logarthmic rule

loga+logb=log(ab)

loga((xy^3)(x^2y))=17

loga(xy)^4=17

4loga(xy)=17

loga(xy)=17/4

Multiply bothsides with 1/2

(1/2)loga(xy)=(1/2)(17/4)

loga(xy)^1/2=17/8

Comments

The value of log_a(xy)^1/2 = 5/2.

Answer 2

logx+3logy=9
2logx+logy=8
logx=a
logy=b
a+3b=9
2a+b=8
solve above equations
b=2 and a=3
1/2(a+b)=1/2(2+3)
1/2(logx+logy)=5/2
log(xy)^1/2=5/2