https://file.aiquickdraw.com/mathbot/file/1726933864824yg8wudyk.jpg

Let's go through the equations shown in the image and solve them step by step.

### 1. **Solve** \( \log_3 (x + 25) - \log_3 (x - 1) = 3 \)

#### Solution:
Using the logarithmic property:
\[
\log_b (A) - \log_b (B) = \log_b \left( \frac{A}{B} \right)
\]
This equation simplifies to:
\[
\log_3 \left( \frac{x + 25}{x - 1} \right) = 3
\]
Now, convert the logarithmic form to its exponential form:
\[
\frac{x + 25}{x - 1} = 3^3 = 27
\]
Multiply both sides by \(x - 1\):
\[
x + 25 = 27(x - 1)
\]
Expand the right side:
\[
x + 25 = 27x - 27
\]
Rearrange the terms:
\[
25 + 27 = 27x - x
\]
\[
52 = 26x
\]
Solve for \(x\):
\[
x = \frac{52}{26} = 2
\]

Thus, \(x = 2\).

---

### 2. **Solve** \( \log_2 (x - 2) + \log_2 (x + 1) = 2 \)

#### Solution:
Using the logarithmic property:
\[
\log_b (A) + \log_b (B) = \log_b (A \cdot B)
\]
This equation becomes:
\[
\log_2 \left( (x - 2)(x + 1) \right) = 2
\]
Convert to exponential form:
\[
(x - 2)(x + 1) = 2^2 = 4
\]
Now expand the left-hand side:
\[
x^2 + x - 2x - 2 = 4
\]
Simplify:
\[
x^2 - x - 2 = 4
\]
Rearrange to form a quadratic equation:
\[
x^2 - x - 6 = 0
\]
Factor the quadratic equation:
\[
(x - 3)(x + 2) = 0
\]
Thus, \(x = 3\) or \(x = -2\).

However, logarithms are not defined for non-positive arguments, so we must reject \(x = -2\), as it leads to negative arguments in the original logarithmic functions.

Therefore, \(x = 3\).

---

### 3. **Solve** \( 4^x = 5^{x + 1} \) using logarithms.

#### Solution:
Take the natural logarithm (or log base 10) of both sides:
\[
\ln(4^x) = \ln(5^{x + 1})
\]
Using logarithmic rules:
\[
x \ln(4) = (x + 1) \ln(5)
\]
Expand the right-hand side:
\[
x \ln(4) = x \ln(5) + \ln(5)
\]
Rearrange to collect the terms involving \(x\):
\[
x \ln(4) - x \ln(5) = \ln(5)
\]
Factor out \(x\):
\[
x (\ln(4) - \ln(5)) = \ln(5)
\]
Solve for \(x\):
\[
x = \frac{\ln(5)}{\ln(4) - \ln(5)}
\]

This is the exact solution. You can calculate the numerical value using logarithms, but this is the simplified form.

---

Do you want further details on any of these solutions?

Here are 5 related questions:
1. How do you handle equations involving logarithms of different bases?
2. What are the properties of logarithms that simplify solving equations?
3. How can you apply logarithms to exponential equations?
4. What are the domain restrictions when solving logarithmic equations?
5. How can you solve quadratic equations that arise from logarithmic problems?

**Tip:** When solving logarithmic equations, always check that the solutions make the arguments of all logarithms positive.

Solve the following logarithmic and exponential equations: log_3(x + 25) - log_3(x - 1) = 3, log_2(x - 2) + log_2(x + 1) = 2, 4^x = 5^(x + 1) (use logarithms).

Learn how to solve logarithmic equations like log_3(x + 25) - log_3(x - 1) = 3, log_2(x - 2) + log_2(x + 1) = 2, and exponential equations such as 4^x = 5^(x + 1). This guide includes key logarithmic and exponential properties, suitable for Grades 10-12 students.

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