Let's analyze the given logarithmic functions and find their domain, range, x-intercepts, and horizontal asymptotes.

1. $f(x) = \log_2 (x + 5)$

• Domain: The argument inside the logarithm, $x + 5$, must be greater than 0. $x + 5 > 0 \quad \Rightarrow \quad x > -5$ So, the domain is $(-5, \infty)$.

• Range: The range of any logarithmic function is $(-\infty, \infty)$.

• X-Intercept: Set $f(x) = 0$ and solve for $x$. $0 = \log_2 (x + 5) \quad \Rightarrow \quad 2^0 = x + 5 \quad \Rightarrow \quad x = -4$ So, the x-intercept is $x = -4$.

• Horizontal Asymptote: Logarithmic functions do not have horizontal asymptotes, so there is none.

2. $f(x) = \log_4 (2x + 5)$

• Domain: The argument inside the logarithm, $2x + 5$, must be greater than 0. $2x + 5 > 0 \quad \Rightarrow \quad x > -\frac{5}{2}$ So, the domain is $\left(-\frac{5}{2}, \infty\right)$.

• Range: The range is $(-\infty, \infty)$.

• X-Intercept: Set $f(x) = 0$ and solve for $x$. $0 = \log_4 (2x + 5) \quad \Rightarrow \quad 4^0 = 2x + 5 \quad \Rightarrow \quad 2x = -4 \quad \Rightarrow \quad x = -2$ So, the x-intercept is $x = -2$.

• Horizontal Asymptote: Logarithmic functions do not have horizontal asymptotes, so there is none.

3. $f(x) = \log_5 (7x - 3)$

• Domain: The argument inside the logarithm, $7x - 3$, must be greater than 0. $7x - 3 > 0 \quad \Rightarrow \quad x > \frac{3}{7}$ So, the domain is $\left(\frac{3}{7}, \infty\right)$.

• Range: The range is $(-\infty, \infty)$.

• X-Intercept: Set $f(x) = 0$ and solve for $x$. $0 = \log_5 (7x - 3) \quad \Rightarrow \quad 5^0 = 7x - 3 \quad \Rightarrow \quad 7x = 4 \quad \Rightarrow \quad x = \frac{4}{7}$ So, the x-intercept is $x = \frac{4}{7}$.

• Horizontal Asymptote: There is no horizontal asymptote.

Let me know if you need more details or have any questions.

Related Questions:

1. How do transformations affect the domain and range of logarithmic functions?
2. Can logarithmic functions ever have horizontal asymptotes? Why or why not?
3. How do we find vertical asymptotes for logarithmic functions?
4. What is the difference between the growth rate of exponential and logarithmic functions?
5. How do logarithmic functions relate to exponential functions?

Tip: Always ensure the argument inside a logarithmic function is strictly positive when determining the domain.