https://file.aiquickdraw.com/mathbot/file/172598307206868tvuwk0.jpg

The problem you're working on involves solving a logarithmic equation. Let's break down the steps shown in the image for clarity:

### Problem: 
Solve the equation:
\[
\log_3(x + 25) - \log_3(x - 1) = 3
\]

### Step 1: Use Logarithmic Properties
You can use the property of logarithms that states:
\[
\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
\]
Applying this to the equation, we get:
\[
\log_3\left(\frac{x + 25}{x - 1}\right) = 3
\]

### Step 2: Rewrite in Exponential Form
We now convert the logarithmic equation into its exponential form. The equation
\[
\log_b(A) = C
\]
is equivalent to
\[
A = b^C
\]
In this case, it becomes:
\[
\frac{x + 25}{x - 1} = 3^3 = 27
\]

### Step 3: Solve for \( x \)
Now, solve the equation:
\[
\frac{x + 25}{x - 1} = 27
\]
Multiply both sides by \( x - 1 \):
\[
x + 25 = 27(x - 1)
\]
Simplify the equation:
\[
x + 25 = 27x - 27
\]
Move all terms involving \( x \) to one side:
\[
25 + 27 = 27x - x
\]
\[
52 = 26x
\]
Solve for \( x \):
\[
x = \frac{52}{26} = 2
\]

### Final Answer:
\[
x = 2
\]

Would you like more details or have any questions about this?

### Related Questions:
1. How can you solve logarithmic equations with different bases?
2. What are the conditions where logarithmic functions are undefined?
3. How can logarithmic and exponential equations be related?
4. How do you solve logarithmic inequalities?
5. What are the properties of logarithms used to simplify complex expressions?

### Tip:
Always check for any restrictions on the domain of the original logarithmic expressions. In this case, \( x - 1 > 0 \) and \( x + 25 > 0 \), so \( x > 1 \).

Learn how to solve the logarithmic equation log_3(x + 25) - log_3(x - 1) = 3 by using properties of logarithms and converting to exponential form. This step-by-step guide covers key algebraic concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation