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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!

Find the domain, range, x-intercept, and horizontal asymptote of the given logarithmic functions: f(x) = log2(x) + 5, f(x) = log4(2x) + 5, and f(x) = log5(7x - 3).

Understand how to determine the domain, range, x-intercept, and horizontal asymptote of logarithmic functions like log2(x), log4(2x), and log5(7x - 3). This guide is suitable for high school students studying logarithms and graph transformations.

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