https://file.aiquickdraw.com/mathbot/file/1728686264391gpvdkuwu.jpg

The given equation is:

\[
\log_5(x + 4) = 1 - \log_5(x + 8)
\]

### Step 1: Combine logarithmic terms
We begin by moving the logarithmic term \(\log_5(x + 8)\) to the left-hand side:

\[
\log_5(x + 4) + \log_5(x + 8) = 1
\]

### Step 2: Apply logarithmic property
Use the property of logarithms:

\[
\log_b(A) + \log_b(B) = \log_b(A \times B)
\]

This simplifies the left-hand side:

\[
\log_5((x + 4)(x + 8)) = 1
\]

### Step 3: Exponentiate both sides
To remove the logarithm, we exponentiate both sides using base 5:

\[
(x + 4)(x + 8) = 5^1
\]

\[
(x + 4)(x + 8) = 5
\]

### Step 4: Expand and simplify
Now, expand the left-hand side:

\[
x^2 + 8x + 4x + 32 = 5
\]

\[
x^2 + 12x + 32 = 5
\]

Subtract 5 from both sides:

\[
x^2 + 12x + 27 = 0
\]

### Step 5: Solve the quadratic equation
Now, solve the quadratic equation:

\[
x^2 + 12x + 27 = 0
\]

Using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 1\), \(b = 12\), and \(c = 27\). Substituting these values:

\[
x = \frac{-12 \pm \sqrt{12^2 - 4(1)(27)}}{2(1)}
\]

\[
x = \frac{-12 \pm \sqrt{144 - 108}}{2}
\]

\[
x = \frac{-12 \pm \sqrt{36}}{2}
\]

\[
x = \frac{-12 \pm 6}{2}
\]

### Step 6: Find the solutions
We have two possible solutions:

\[
x_1 = \frac{-12 + 6}{2} = \frac{-6}{2} = -3
\]

\[
x_2 = \frac{-12 - 6}{2} = \frac{-18}{2} = -9
\]

### Step 7: Check for extraneous solutions
For logarithms to be valid, the arguments inside the logarithms must be positive. Therefore, we check the conditions:

- For \(x = -3\), \(\log_5(x + 4) = \log_5(1)\) and \(\log_5(x + 8) = \log_5(5)\), both are valid.
- For \(x = -9\), \(\log_5(x + 4) = \log_5(-5)\), which is undefined (since logarithms of negative numbers are not allowed).

Thus, \(x = -9\) is an extraneous solution.

### Final Solution:
The only valid solution is:

\[
x = -3
\]

Would you like further clarification or more details on any step?

Here are 5 related questions:
1. How do you apply the properties of logarithms when combining terms?
2. What is the importance of checking for extraneous solutions in logarithmic equations?
3. How can you solve quadratic equations by factoring instead of using the quadratic formula?
4. What other methods can be used to solve logarithmic equations?
5. How does changing the base of a logarithm affect the equation?

**Tip:** Always check the domain of the logarithmic function to ensure the arguments are valid before finalizing your solution.

Solve for x: log5(x + 4) = 1 - log5(x + 8).

Learn how to solve the logarithmic equation log5(x + 4) = 1 - log5(x + 8), which simplifies to a quadratic equation. This step-by-step guide covers key logarithmic properties and the quadratic formula, 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