Solve the following equation.

log7(*x* + 5) − log7(*x* − 1) = 1

We are given the logarithmic equation:

\[
\log_7(x + 5) - \log_7(x - 1) = 1
\]

### Step 1: Use the logarithm subtraction rule.

The logarithm subtraction rule states that:

\[
\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
\]

Apply this to the given equation:

\[
\log_7\left(\frac{x + 5}{x - 1}\right) = 1
\]

### Step 2: Rewrite the equation in exponential form.

Since \( \log_b(A) = C \) implies \( A = b^C \), we can rewrite the equation as:

\[
\frac{x + 5}{x - 1} = 7^1
\]

Simplifying:

\[
\frac{x + 5}{x - 1} = 7
\]

### Step 3: Solve for \(x\).

Multiply both sides by \(x - 1\) to clear the fraction:

\[
x + 5 = 7(x - 1)
\]

Expand and solve for \(x\):

\[
x + 5 = 7x - 7
\]

Subtract \(x\) from both sides:

\[
5 = 6x - 7
\]

Add 7 to both sides:

\[
12 = 6x
\]

Divide by 6:

\[
x = 2
\]

### Step 4: Verify the solution.

Substitute \(x = 2\) back into the original logarithmic expressions to ensure they are valid:

\[
\log_7(2 + 5) - \log_7(2 - 1) = \log_7(7) - \log_7(1)
\]

This simplifies to:

\[
\log_7(7) - \log_7(1) = 1 - 0 = 1
\]

The solution is valid.

### Final Answer:
The solution to the equation \( \log_7(x + 5) - \log_7(x - 1) = 1 \) is:

\[
x = 2
\]

Would you like more details or have any questions? Here are some related problems to practice:

1. Solve \(\log_2(x + 3) - \log_2(x - 1) = 2\).
2. Solve \(\log_5(3x + 1) = 2\).
3. Solve \(\log_{10}(x + 4) = 2 + \log_{10}(x)\).
4. Solve \(\log_4(x^2 - 4) = 1\).
5. Solve \(2\log_3(x) = \log_3(9)\).

**Tip:** Always check the domain of the logarithmic expressions to ensure the argument inside the logarithm is positive.

Learn how to solve the logarithmic equation log7(x + 5) - log7(x - 1) = 1 with a step-by-step solution. This problem involves logarithmic functions and equations, using the logarithm subtraction rule to simplify and solve. Suitable for students in Grades 10-12.

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