Use the Laws of logarithms to rewrite the expression

log3(x^18*sqrt3(y12))
in a form with no logarithm of a product, quotient or power.
After rewriting we have
log3(x^18*sqrt3(y12))=Alog3(x)+Blog3(y)
With the constant A=? B=?
I have no idea with this one and my teacher is so busy so I can't ask.
Please show me steps, Tks very much

np

To rewrite the expression log3(x^18 * sqrt3(y^12)) without logarithm of a product, quotient, or power, we can use the laws of logarithms.

The first law we can use is the power law of logarithms, which states that log base a of (x^y) is equal to y times log base a of x.
In this case, we can rewrite x^18 as 18 * log3(x).

Next, we can use the property of square root as exponent. So √y^12 is the same as y^(12/2) = y^6.

Now we can rewrite the expression:
log3(x^18 * sqrt3(y^12))
= log3(x^18 * y^6)

To rewrite the expression in the form Alog3(x) + Blog3(y), we need to separate the terms involving x and y.

Using the multiplication law of logarithms, we can rewrite the expression as:
log3(x^18) + log3(y^6)

Using the power law of logarithms on each term, we get:
18 * log3(x) + 6 * log3(y)

So, A = 18 and B = 6.

Therefore, the rewritten expression is:
log3(x^18 * sqrt3(y^12)) = 18 * log3(x) + 6 * log3(y)