Find x in the following equation.
log Subscript 10 Baseline left parenthesis x plus 2 right parenthesis minus log Subscript 10 Baseline left parenthesis x minus 2 right parenthesis equals 1
Question content area bottom
Part 1
x equals
  
enter your response here ​(Type a fraction or an integer. Simplify your​ answer.)

The given equation is:

\[
\log_{10}(x + 2) - \log_{10}(x - 2) = 1
\]

### Step 1: Apply the properties of logarithms

Use the logarithmic identity:

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

This simplifies the equation as follows:

\[
\log_{10}\left(\frac{x + 2}{x - 2}\right) = 1
\]

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

To eliminate the logarithm, we rewrite it using the definition of a logarithm:

\[
\log_b(y) = z \quad \text{implies} \quad y = b^z
\]

So, we have:

\[
\frac{x + 2}{x - 2} = 10^1 = 10
\]

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

Now, solve the equation:

\[
\frac{x + 2}{x - 2} = 10
\]

Multiply both sides by \(x - 2\) to get rid of the denominator:

\[
x + 2 = 10(x - 2)
\]

Expand the right-hand side:

\[
x + 2 = 10x - 20
\]

Move all terms involving \(x\) to one side and constant terms to the other side:

\[
2 + 20 = 10x - x
\]

\[
22 = 9x
\]

Solve for \(x\):

\[
x = \frac{22}{9}
\]

Thus, the solution is:

\[
x = \frac{22}{9}
\]

### Would you like further clarification or details on any part of the solution?

---

Here are some follow-up questions related to the topic:

1. What happens if the logarithmic base is changed in the original equation?
2. How do you handle equations with multiple logarithmic terms?
3. What are the properties of logarithms that can help simplify complex expressions?
4. What other types of equations can be solved using logarithmic properties?
5. Can you solve similar equations with natural logarithms (\(\ln\)) instead of common logarithms (\(\log_{10}\))?

**Tip:** Always check for domain restrictions in logarithmic equations since the arguments of the logarithms must be positive!

Find x in the following equation: log_10(x + 2) - log_10(x - 2) = 1. (Enter your response as a fraction or integer)

Learn how to solve the logarithmic equation log_10(x + 2) - log_10(x - 2) = 1 step-by-step. This problem covers key logarithmic concepts, properties of logarithms, and how to use exponential form to find the solution. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation