The problem you're working on involves solving a logarithmic equation. Let's break down the steps shown in the image for clarity:

Problem:

Solve the equation: $\log_3(x + 25) - \log_3(x - 1) = 3$

Step 1: Use Logarithmic Properties

You can use the property of logarithms that states: $\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$ Applying this to the equation, we get: $\log_3\left(\frac{x + 25}{x - 1}\right) = 3$

Step 2: Rewrite in Exponential Form

We now convert the logarithmic equation into its exponential form. The equation $\log_b(A) = C$ is equivalent to $A = b^C$ In this case, it becomes: $\frac{x + 25}{x - 1} = 3^3 = 27$

Step 3: Solve for $x$

Now, solve the equation: $\frac{x + 25}{x - 1} = 27$ Multiply both sides by $x - 1$: $x + 25 = 27(x - 1)$ Simplify the equation: $x + 25 = 27x - 27$ Move all terms involving $x$ to one side: $25 + 27 = 27x - x$ $52 = 26x$ Solve for $x$: $x = \frac{52}{26} = 2$

Final Answer:

$x = 2$

Would you like more details or have any questions about this?

Related Questions:

1. How can you solve logarithmic equations with different bases?
2. What are the conditions where logarithmic functions are undefined?
3. How can logarithmic and exponential equations be related?
4. How do you solve logarithmic inequalities?
5. What are the properties of logarithms used to simplify complex expressions?

Tip:

Always check for any restrictions on the domain of the original logarithmic expressions. In this case, $x - 1 > 0$ and $x + 25 > 0$, so $x > 1$.