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### 1) Domain and Image of the Functions
   - **\( f(x) = \log_2(x-2) \)**:
     - **Domain**: For the logarithmic function to be defined, the argument must be greater than 0. So, \( x - 2 > 0 \), or \( x > 2 \).
     - **Image**: The range of a logarithmic function \( \log_2(x - 2) \) is \( (-\infty, \infty) \).
     
   - **\( g(x) = \log(x + 2) \)**:
     - **Domain**: For the logarithmic function to be defined, \( x + 2 > 0 \), or \( x > -2 \).
     - **Image**: Similar to the previous logarithmic function, the range is \( (-\infty, \infty) \).

### 2) Graphical Representation of Logarithmic Functions
   - **a)** How are the graphs based on the given logarithmic bases?
     - The base of a logarithm affects the steepness of the graph. A larger base makes the graph grow more slowly. The general shape for both functions is increasing, but the base of 2 in \( f(x) = \log_2(x-2) \) will grow slower compared to the natural logarithm.
   - **b)** How do the logarithmic functions' graphs behave?
     - Both functions will increase as \( x \) increases, and they will have vertical asymptotes at \( x = 2 \) for \( f(x) \) and \( x = -2 \) for \( g(x) \).
   - **c)** At what point of the x-axis do they intersect?
     - To find the intersection, solve \( \log_2(x-2) = \log(x+2) \). This will give the x-value where the graphs intersect.

### 3) Solve the Exponential Equation
   - \( 3^{x+3} = 9 \):
     - We rewrite \( 9 \) as \( 3^2 \). So, \( 3^{x+3} = 3^2 \).
     - Equating exponents, \( x + 3 = 2 \), thus \( x = -1 \).
     - Correct answer: **(c) -1**.

### 4) Logarithmic Equation
   - \( \log_3 x - \log_3 (x - 2) = 2 \):
     - Apply the properties of logarithms: \( \log_3 \left( \frac{x}{x-2} \right) = 2 \).
     - This implies \( \frac{x}{x-2} = 3^2 = 9 \).
     - Solve for \( x \): \( x = 9(x - 2) \Rightarrow x = 9x - 18 \Rightarrow 8x = 18 \Rightarrow x = \frac{9}{4} \).
     - The correct answer is not explicitly listed, but this equation simplifies further from option **b)**.

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Do you need further clarification on any of the steps?

Here are 5 additional questions to explore:

1. How do you solve logarithmic equations with different bases?
2. How do exponential functions behave compared to logarithmic functions?
3. How can you derive the range of a function from its graph?
4. What happens when the base of a logarithmic function is less than 1?
5. How do you solve logarithmic inequalities?

**Tip**: Always check the domain restrictions when solving logarithmic equations to ensure that your solution is valid.

Indicate the domain and range of the following functions: f(x) = log2(x - 2), g(x) = log(x + 2). Also, solve a logarithmic equation and solve an exponential equation.

Explore how to determine the domain and range of logarithmic functions such as f(x) = log2(x - 2) and g(x) = log(x + 2), solve exponential and logarithmic equations, and understand the properties of logarithms. This is suitable for Grades 10-12 students learning advanced algebra.

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