Let's go through these problems step by step.

Part (a):

$\ln(x + 4) - \ln(x - 4)$

Using the logarithmic property: $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ We can apply this property here: $\ln(x + 4) - \ln(x - 4) = \ln\left(\frac{x + 4}{x - 4}\right)$ Thus, the expression simplifies to: $\ln\left(\frac{x + 4}{x - 4}\right)$

Part (b):

$\ln(x + 4) + \ln(x - 4)$

Using the logarithmic property: $\ln(a) + \ln(b) = \ln(a \cdot b)$ We apply this property here: $\ln(x + 4) + \ln(x - 4) = \ln\left((x + 4)(x - 4)\right)$ Since $(x + 4)(x - 4)$ is a difference of squares, we can further simplify: $(x + 4)(x - 4) = x^2 - 16$ Thus, the expression simplifies to: $\ln(x^2 - 16)$

Final answers:

• (a): $\ln\left(\frac{x + 4}{x - 4}\right)$
• (b): $\ln(x^2 - 16)$

Would you like any more details or clarifications on these solutions?

Relative Questions:

1. What are other logarithmic properties, such as for multiplication and division?
2. How does the natural logarithm differ from other logarithms?
3. Can you apply these properties to expressions with more than two logarithmic terms?
4. What is the inverse of the natural logarithmic function?
5. How would the logarithmic properties apply to logarithms with different bases?

Tip:

When dealing with logarithmic expressions, remember that the subtraction and addition of logs correspond to division and multiplication, respectively. This helps simplify complex expressions.