We are given the equation:

\[
\log_3 (x + 25) - \log_3 (x - 1) = 3
\]

We can use the properties of logarithms to simplify this expression. Specifically, the difference of logarithms can be written as the logarithm of a quotient:

\[
\log_3 \left( \frac{x + 25}{x - 1} \right) = 3
\]

Next, rewrite the logarithmic equation in exponential form. The equation \(\log_b(y) = z\) is equivalent to \(b^z = y\). In our case, \(b = 3\), so:

\[
\frac{x + 25}{x - 1} = 3^3
\]

Simplifying:

\[
\frac{x + 25}{x - 1} = 27
\]

Now, multiply both sides by \((x - 1)\) to eliminate the denominator:

\[
x + 25 = 27(x - 1)
\]

Expand the right-hand side:

\[
x + 25 = 27x - 27
\]

Now, collect like terms:

\[
25 + 27 = 27x - x
\]

\[
52 = 26x
\]

Solve for \(x\):

\[
x = \frac{52}{26} = 2
\]

Now, let's check if \(x = 2\) satisfies the original equation. Substitute \(x = 2\) into the logarithmic terms:

\[
\log_3 (2 + 25) - \log_3 (2 - 1) = \log_3 (27) - \log_3 (1)
\]

\[
\log_3 (27) = 3 \quad \text{and} \quad \log_3 (1) = 0
\]

So:

\[
3 - 0 = 3
\]

This confirms that \(x = 2\) is the correct solution.

The solution is:

\[
x = 2
\]

Would you like more details on any step? Here are some related questions you might find helpful:

1. How do we derive logarithmic properties?
2. What is the general process for solving logarithmic equations?
3. How do you convert logarithmic form to exponential form?
4. How can we solve equations involving logarithms of different bases?
5. What are the key characteristics of logarithmic functions?

**Tip:** When solving logarithmic equations, always check if the solution satisfies the domain restrictions of the logarithms (arguments must be positive).

This guide shows how to solve the logarithmic equation log3(x + 25) − log3(x − 1) = 3. Learn key logarithmic properties and how to convert logarithmic form to exponential form. Perfect for high school students, Grades 9-12.

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