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

Steps to Solve

1. Apply the power rule to the second term

Using the power rule of logarithms, $a \ln(b) = \ln(b^a)$, we can rewrite the second term: $$ 2 \ln(x - 4) = \ln((x - 4)^2) $$

2. Combine the logarithmic terms

Now substitute the rewritten term back into the expression: $$ \ln(x^2 - 13x + 36) + \ln((x - 4)^2) - \ln(x - 9) $$ Using the property $\ln(a) + \ln(b) = \ln(ab)$, we combine the first two logarithmic terms: $$ \ln((x^2 - 13x + 36)(x - 4)^2) $$

3. Apply the quotient rule

Now, apply the quotient rule for logarithms, $\ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right)$: $$ \ln\left(\frac{(x^2 - 13x + 36)(x - 4)^2}{x - 9}\right) $$

4. Simplifying the expression

Next, we need to factor $x^2 - 13x + 36$ to combine or simplify further. The factors of this quadratic are $(x - 4)(x - 9)$. Thus, we can rewrite our expression: $$ \ln\left(\frac{(x - 4)(x - 9)(x - 4)^2}{x - 9}\right) $$

5. Final simplification

Simplifying the fraction, the $x - 9$ in the numerator and denominator cancels out: $$ \ln((x - 4)^3) $$