The image contains a problem that asks to solve for $x$. Here's what I can extract from the image:

Problem 4: Solve for $x$

$a) \log_3{x} = \log_3{4} + \log_3{5}$

$b) (\log_3{x} - 8) = 1$

Let's solve both equations.

Part a) $\log_3{x} = \log_3{4} + \log_3{5}$

1. Using the property of logarithms that states:
   $\log_b{A} + \log_b{B} = \log_b{(A \cdot B)}$ we can simplify the right-hand side: $\log_3{x} = \log_3{(4 \cdot 5)}$ $\log_3{x} = \log_3{20}$

2. Since the logarithms on both sides have the same base, we can equate the arguments: $x = 20$

Part b) $\log_3{x} - 8 = 1$

1. Add 8 to both sides: $\log_3{x} = 9$

2. Using the definition of logarithms, we convert this into an exponential equation: $x = 3^9$ $x = 19683$

Solutions:

• Part a: $x = 20$
• Part b: $x = 19683$

Would you like more details, or do you have any further questions?

Here are some follow-up questions to explore the topic further:

1. What are other logarithmic properties, and how are they used in solving equations?
2. How would the solution change if the logarithmic bases were different?
3. Can you apply the logarithmic change of base formula to these equations?
4. How does the logarithmic function behave as $x$ approaches infinity?
5. How can logarithmic equations be used in real-world scenarios?

Tip: Always check if you can simplify logarithmic expressions by combining them before solving the equation.