Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. Assume that the variables represent positive real numbers. log3(1/9a^3b^2)

Ln( (1/9)*a^3*b^2) = 3*Ln(a) + 2*Ln(b) - Ln(9)

Step-by-step explanation:

Here we need to use the relations:

Ln(x^y) = y*Ln(x)

Ln(x*y) = Ln(x) + Ln(y)

Ln(x/y) = Ln(x) - Ln(y)

Then, we have:

Ln( (1/9)*a^3*b^2)

First we could use the second relation to rewrite this as:

Ln(1/9) + Ln(a^3) + Ln(b^2)

Now we can use the third relationship to rewrite the first term as:

ln(1/9) = ln(1) - ln(9) = -ln(9)

Then our equation becomes:

Ln(1/9) + Ln(a^3) + Ln(b^2) = Ln(a^3) + Ln(b^2) - ln(9)

Now we can use the first relation to rewrite this as:

Ln(a^3) + Ln(b^2) - ln(9) = 3*Ln(a) + 2*Ln(b) - Ln(9)

And that is all the simplifications we can do.