ln(x^2 - 13x + 36) + 2ln(x - 4) - ln(x - 9)

Steps to Solve

1. Apply the Property of Logarithms for Multiplication

Using the property of logarithms that states $a \ln(b) = \ln(b^a)$, we can simplify $2 \ln(x - 4)$:

$$ 2 \ln(x - 4) = \ln((x - 4)^2) $$

Now the expression becomes: $$ \ln(x^2 - 13x + 36) + \ln((x - 4)^2) - \ln(x - 9) $$

2. Combine the Logarithmic Expressions

Next, we can apply the property of logarithms for addition which states that $\ln(a) + \ln(b) = \ln(ab)$:

$$ \ln(x^2 - 13x + 36) + \ln((x - 4)^2) = \ln((x^2 - 13x + 36)(x - 4)^2) $$

So now our expression is: $$ \ln((x^2 - 13x + 36)(x - 4)^2) - \ln(x - 9) $$

3. Apply the Property of Logarithms for Subtraction

Using the property of logarithms for subtraction, which states that $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$, we can write:

$$ \ln\left(\frac{(x^2 - 13x + 36)(x - 4)^2}{(x - 9)}\right) $$

4. Factor the Quadratic Expression

Now, let's factor $x^2 - 13x + 36$: $$ x^2 - 13x + 36 = (x - 4)(x - 9) $$

Substituting this back into our expression, we get: $$ \ln\left(\frac{(x - 4)(x - 9)(x - 4)^2}{x - 9}\right) $$

5. Simplify the Expression

The expression simplifies as follows: $$ \ln\left((x - 4)^3\right) $$

So the final simplified expression is: $$ \ln((x - 4)^3) $$