https://file.aiquickdraw.com/mathbot/file/1726575023830xr6h1h64.jpg

The problem given is:

\[
\log_5 (7t + 2) - \log_5 t = 4
\]

### Step-by-Step Solution:

1. **Apply the logarithmic subtraction rule**:
   The difference of logarithms can be written as the logarithm of a quotient:
   \[
   \log_5 \left( \frac{7t + 2}{t} \right) = 4
   \]

2. **Rewrite the equation in exponential form**:
   Since \( \log_b (x) = y \) implies \( b^y = x \), we can rewrite the equation as:
   \[
   \frac{7t + 2}{t} = 5^4
   \]

   Now calculate \( 5^4 \):
   \[
   5^4 = 625
   \]

   Thus, the equation becomes:
   \[
   \frac{7t + 2}{t} = 625
   \]

3. **Solve for \( t \)**:
   Multiply both sides of the equation by \( t \) to eliminate the denominator:
   \[
   7t + 2 = 625t
   \]

   Rearrange the equation:
   \[
   7t - 625t = -2
   \]
   \[
   -618t = -2
   \]

   Divide both sides by \( -618 \):
   \[
   t = \frac{-2}{-618} = \frac{1}{309}
   \]

Thus, the value of \( t \) is:

\[
t = \frac{1}{309}
\]

---

Would you like more details on logarithmic properties or any other steps?

### Related Questions:
1. What is the logarithmic quotient rule, and how is it applied in equations?
2. How do you convert logarithmic equations into exponential form?
3. How do you solve equations with logarithmic terms on both sides?
4. What are common mistakes to avoid when solving logarithmic equations?
5. Can you apply similar logarithmic properties to different bases other than 5?

**Tip:** When solving logarithmic equations, always ensure the argument of the logarithm is positive for real solutions.

Find the value of t that satisfies the following equation: log_5 (7t + 2) - log_5 t = 4.

This problem demonstrates how to solve the logarithmic equation log_5 (7t + 2) - log_5 t = 4. Using logarithmic properties and the exponential form, the solution involves simplifying and solving for t. Suitable for students in Grades 10-12.

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