We have the logarithmic equation:

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

### Step 1: Apply the logarithmic property
Use the property of logarithms:
\[
\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)
\]

So the equation becomes:
\[
\log_3\left(\frac{x + 25}{x - 1}\right) = 3
\]

### Step 2: Convert to exponential form
Using the fact that \(\log_b(y) = z\) implies \(b^z = y\), we rewrite the equation as:
\[
\frac{x + 25}{x - 1} = 3^3
\]
\[
\frac{x + 25}{x - 1} = 27
\]

### Step 3: Solve the equation
Multiply both sides by \(x - 1\) to eliminate the fraction:
\[
x + 25 = 27(x - 1)
\]
Now, expand and solve for \(x\):
\[
x + 25 = 27x - 27
\]
\[
25 + 27 = 27x - x
\]
\[
52 = 26x
\]
\[
x = \frac{52}{26} = 2
\]

### Step 4: Check for extraneous solutions
The solution \(x = 2\) must satisfy the condition that the arguments of the logarithms are positive:
\[
x + 25 = 2 + 25 = 27 \quad \text{(positive)}
\]
\[
x - 1 = 2

Step-by-step solution to the logarithmic equation log3(x + 25) - log3(x - 1) = 3. Learn how to apply logarithmic properties and convert to exponential form to find the solution. This problem is ideal for students in Grades 10-12.

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