can someone help me solve this logarithm expression?

Question

combine into a single logarithm and siplify
4logx- log(x^2y)+ log(3y)

they are all base 10..
any ideas on how to go about this one? is the only problem i couldnt solve!!!

Answer 1

A few rules that are important:

k log(x) = log(x^k)
log(a) + log(b) = log(ab)
log(a) - log(b) = log(a/b)

Using the first rule on the first log, bring the 4 inside the parentheses as a power:
log(x^4) - log(x²y) + log(3y)

Using the second rule, combine the first two logs:
log(x^4 / x²y) + log(3y)

Simplify the expression inside the first log:
log(x² / y) + log(3y)

Using the 3rd rule, combine these two logs:
log((x² / y) * 3y)

Again simplify by canceling y in the denominator and numerator:
log(x² * 3)

This becomes:
log(3x²)

Answer 2

4logx- log(x^2y)+ log(3y)
= log[x^4*3y/(x^2 y)]
= log(3x^2)

Answer 3

4logx = log (x^4)
log x - log y = log (x/y)
log x + log y = log (xy)
with these rules
4logx- log(x^2y)+ log(3y) =
log[x^4(3y)/(x^2y)] =
log[3x^2]