log (sqrt4(x^16 / y^4))

Question

What is the value of the expression when logx=8 and logy=1?

My work:
x=10^8 y=10

I tried to plug these in but my calculator but it said "overflow." Is there an easier way to do this?

Answer 1

I assume sqrt4(n) means 4th root. So,

log 4throot(x^16/y^4) = log(x^4/y)
= 4logx - logy
= 4*8 - 1
= 31

Of course, you could have worked it without extracting the 4th root:

log 4throot(x^16/y^4)
= 1/4 log(x^16/y^4)
= 1/4 (16logx - 4logy)
= 1/4 (16*8-4*1)
= 1/4 (128-4)
= 1/4 * 124
= 31