3. $$ \quad \log _{5}(x+1)-\log _{5} x=2 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\log_{5}(x+1) - \log_{5}(x) = 2
$$

**Step 1: Apply Logarithm Properties**

Recall the property of logarithms that states:

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

Applying this to the given equation:

$$
\log_{5}\left(\frac{x+1}{x}\right) = 2
$$

**Step 2: Convert to Exponential Form**

The equation $$\log_{5}(y) = 2$$ is equivalent to $$y = 5^2$$. Applying this:

$$
\frac{x+1}{x} = 25
$$

**Step 3: Solve for $$x$$**

Multiply both sides by $$x$$ (noting that $$x > 0$$ since it's inside a logarithm):

$$
x + 1 = 25x
$$

Subtract $$x$$ from both sides:

$$
1 = 24x
$$

Divide both sides by 24:

$$
x = \frac{1}{24}
$$

**Step 4: Verify the Solution**

Ensure that the arguments of the logarithms are positive:

- $$x = \frac{1}{24} > 0$$
- $$x + 1 = \frac{25}{24} > 0$$

Both conditions are satisfied.

**Final Answer:**

$$
x = \frac{1}{24}
$$
$$ x = \frac{1}{24} $$

3. \( \quad \log _{5}(x+1)-\log _{5} x=2 \)