Sure, we can combine and simplify the given expression using properties of logarithms.
We start with the expression:
[tex]\[
\log\left(x^2 + 10x + 24\right) - \log\left(x^2 - 36\right)
\][/tex]
### Step 1: Apply the Logarithm Property
Use the property of logarithms that states [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex]:
[tex]\[
\log\left(x^2 + 10x + 24\right) - \log\left(x^2 - 36\right) = \log\left( \frac{x^2 + 10x + 24}{x^2 - 36} \right)
\][/tex]
### Step 2: Factor the Quadratic Expressions (Optional)
We can factor the quadratic expressions to simplify further, but it is not necessary if we only need a single logarithm.
Factor [tex]\(x^2 + 10x + 24\)[/tex]:
[tex]\[ x^2 + 10x + 24 = (x + 4)(x + 6) \][/tex]
Factor [tex]\(x^2 - 36\)[/tex] which is a difference of squares:
[tex]\[ x^2 - 36 = (x + 6)(x - 6) \][/tex]
### Step 3: Simplify the Fraction (Optional Simplification)
Given these factorizations, we can simplify the fraction inside the logarithm:
[tex]\[
\frac{x^2 + 10x + 24}{x^2 - 36} = \frac{(x + 4)(x + 6)}{(x + 6)(x - 6)}
\][/tex]
Cancel out the common factor [tex]\((x + 6)\)[/tex]:
[tex]\[
\frac{(x + 4)(x + 6)}{(x + 6)(x - 6)} = \frac{x + 4}{x - 6}
\][/tex]
Thus:
[tex]\[
\log\left( \frac{x^2 + 10x + 24}{x^2 - 36} \right) = \log\left( \frac{x + 4}{x - 6} \right)
\][/tex]
### Final Simplified Answer
So, the simplified form of the given logarithmic expression is:
[tex]\[
\log\left(x^2 + 10x + 24\right) - \log\left(x^2 - 36\right) = \log\left( \frac{x^2 + 10x + 24}{x^2 - 36} \right)
\][/tex]
If simplifying further as shown in the optional steps:
[tex]\[
\log\left( \frac{x^2 + 10x + 24}{x^2 - 36} \right) = \log\left( \frac{x + 4}{x - 6} \right)
\][/tex]
So your final answer is:
[tex]\[
\log\left( \frac{x + 4}{x - 6} \right)
\][/tex]
