You have a table with three logarithmic functions and columns to fill in the Domain, Range, x-intercept, and Horizontal Asymptote (H.A.). Let's go over each function and determine these properties:

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

• Domain: The argument inside the logarithm must be greater than zero. Therefore, $x > 0$.
  Domain: $(0, \infty)$

• Range: The range of any logarithmic function is all real numbers. Adding 5 shifts the function vertically but does not change the range.
  Range: $(-\infty, \infty)$

• x-intercept: Set $f(x) = 0$ to find the x-intercept:
  $0 = \log_2(x) + 5 \implies \log_2(x) = -5 \implies x = 2^{-5} = \frac{1}{32}$ x-intercept: $x = \frac{1}{32}$

• Horizontal Asymptote (H.A.): A logarithmic function approaches $-\infty$ as $x$ approaches 0 from the right. The vertical shift by 5 does not affect the horizontal asymptote.
  H.A.: $y = 5$

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

• Domain: The argument inside the logarithm must be positive:
  $2x > 0 \implies x > 0$.
  Domain: $(0, \infty)$

• Range: The range of the logarithmic function is still all real numbers, unaffected by the 5.
  Range: $(-\infty, \infty)$

• x-intercept: Set $f(x) = 0$ to find the x-intercept:
  $0 = \log_4(2x) + 5 \implies \log_4(2x) = -5 \implies 2x = 4^{-5} = \frac{1}{1024} \implies x = \frac{1}{2048}$ x-intercept: $x = \frac{1}{2048}$

• Horizontal Asymptote (H.A.): The same reasoning applies as in the first function. The vertical shift of 5 moves the horizontal asymptote to $y = 5$.
  H.A.: $y = 5$

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

• Domain: Set the argument inside the logarithm greater than zero:
  $7x - 3 > 0 \implies x > \frac{3}{7}$.
  Domain: $\left( \frac{3}{7}, \infty \right)$

• Range: The range is still all real numbers.
  Range: $(-\infty, \infty)$

• x-intercept: Set $f(x) = 0$ to find the x-intercept:
  $0 = \log_5(7x - 3) \implies 7x - 3 = 1 \implies 7x = 4 \implies x = \frac{4}{7}$ x-intercept: $x = \frac{4}{7}$

• Horizontal Asymptote (H.A.): For this function, the horizontal asymptote corresponds to the logarithmic behavior as $7x - 3 \to 0^+$, which happens as $x \to \frac{3}{7}$ from the right.
  H.A.: No horizontal asymptote for this one, since logarithmic functions typically only have vertical asymptotes based on their domain.

Would you like further clarification or details on these properties?

Here are 5 related questions to explore:

1. How do vertical and horizontal shifts affect the graph of a logarithmic function?
2. What happens to the domain if you have a negative coefficient inside the logarithm?
3. How do you graph a transformed logarithmic function like $\log_2(x) + 5$?
4. What is the general process for solving $\log_b(x) = y$?
5. Can a logarithmic function have more than one x-intercept?

Tip: When working with logarithmic functions, always start by analyzing the domain carefully to ensure you avoid non-real arguments!