Let's break down the two parts of the math problem from the image:

Part I:

1. Write the equivalent logarithmic function:

Given the expression: $\log_3(x+1) - 2\log_3(x+1) + 3\log_3x$ You can simplify using the logarithmic properties:

• The logarithmic subtraction rule: $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$
• The logarithmic multiplication rule: $a\log_b(c) = \log_b(c^a)$

2. Solve the equation:

$\log(2x) - \log(3x) = -1$ You will use the logarithmic subtraction property $\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)$, then convert it into exponential form and solve for $x$.

Part II:

Find a real-life problem involving exponential or logarithmic functions, solve it, and box the final answer.

Let's begin solving each part:

Part I:

Step 1: Simplify the logarithmic expression

$\log_3(x+1) - 2\log_3(x+1) + 3\log_3(x)$ Simplifying the terms: $\log_3(x+1) - 2\log_3(x+1) = \log_3(x+1) - \log_3((x+1)^2) = \log_3\left(\frac{x+1}{(x+1)^2}\right)$ Further simplifying: $\log_3\left(\frac{1}{x+1}\right) + 3\log_3(x) = \log_3\left(\frac{1}{x+1}\right) + \log_3(x^3)$ Using the logarithmic addition rule: $\log_3\left(\frac{x^3}{x+1}\right)$

Thus, the simplified logarithmic function is: $\log_3\left(\frac{x^3}{x+1}\right)$

Step 2: Solve the equation $\log(2x) - \log(3x) = -1$

Using the property of logarithms: $\log\left(\frac{2x}{3x}\right) = -1$ Simplifying the fraction: $\log\left(\frac{2}{3}\right) = -1$ Rewrite this in exponential form: $\frac{2}{3} = 10^{-1}$ Solving for the numerical values: $\frac{2}{3} = 0.1$ This is not possible, indicating no solution exists for this equation.

Would you like to go into more details about the solutions?

Here are five related questions to expand on this topic:

1. What is the change of base formula for logarithms, and how can it be applied?
2. Can logarithmic functions have negative arguments or results?
3. How can you check if your solution to a logarithmic equation is correct?
4. What real-life problems can be modeled using logarithmic functions?
5. How do exponential and logarithmic functions relate to each other?

Tip: When solving logarithmic equations, always check if the argument of the logarithm is positive, as logarithms are only defined for positive values of the argument.