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=?
To rewrite the expression log3(x^18*sqrt3(y^12)) in a form without logarithm of a product, quotient, or power, we can apply the laws of logarithms.
The laws of logarithms state:
1. log(a * b) = log(a) + log(b)
2. log(a / b) = log(a) - log(b)
3. log(a^c) = c * log(a)
First, let's break down the expression:
log3(x^18 * sqrt3(y^12))
1. Using the power rule, we can rewrite x^18 as (x^2)^9. Similarly, we can rewrite sqrt3(y^12) as (y^6)^(1/2).
So, the expression becomes log3((x^2)^9 * (y^6)^(1/2)).
2. Using the product rule, we can rewrite the expression as the sum of logarithms:
log3((x^2)^9 * (y^6)^(1/2)) = 9 * log3(x^2) + (1/2) * log3(y^6).
Now, comparing this to the desired form A * log3(x) + B * log3(y), we can find the values of A and B.
In our expression, A = 9 and B = (1/2).
So, the rewritten expression is:
log3(x^18 * sqrt3(y^12)) = 9 * log3(x^2) + (1/2) * log3(y^6)
With A = 9 and B = 1/2.