The expression in the image is:
$\log_5(x^2 - 5x + 4)$

Let’s simplify and analyze this logarithmic expression:

Step 1: Factorize the quadratic expression

The quadratic expression $x^2 - 5x + 4$ can be factorized as: $x^2 - 5x + 4 = (x - 4)(x - 1)$

Step 2: Rewrite the logarithmic expression

Substituting the factorization into the logarithmic expression: $\log_5(x^2 - 5x + 4) = \log_5((x - 4)(x - 1))$

Step 3: Apply the logarithmic property

Using the logarithmic property $\log_b(mn) = \log_b(m) + \log_b(n)$, we can rewrite this as: $\log_5((x - 4)(x - 1)) = \log_5(x - 4) + \log_5(x - 1)$

Final Answer:

$\log_5(x^2 - 5x + 4) = \log_5(x - 4) + \log_5(x - 1)$

Domain Consideration:

For the logarithmic expression to be valid, the argument must be positive. This means: $x^2 - 5x + 4 > 0 \quad \text{or} \quad (x - 4)(x - 1) > 0$ Using the sign chart:

• The solution is $x \in (-\infty, 1) \cup (4, \infty)$, where the product is positive.

Do you want further clarification or examples?

Related Questions:

1. What is the base-5 logarithmic property for division?
2. How do you factorize a quadratic expression like $x^2 - 5x + 4$?
3. How do we determine the domain of a logarithmic function?
4. Can we graph the function $\log_5(x^2 - 5x + 4)$?
5. How is the sign chart constructed for inequalities involving factors?

Tip:

Always check the domain of a logarithmic function to avoid undefined values caused by negative arguments or zero.