**Solve the following logarithmic equation:log7x + log7(3x-2) = 0**

We are given the equation:

\[
\log_7{x} + \log_7{(3x - 2)} = 0
\]

### Step 1: Apply the Logarithmic Property
We can use the logarithmic property:

\[
\log_b{A} + \log_b{B} = \log_b{(A \cdot B)}
\]

Thus, the equation becomes:

\[
\log_7{(x \cdot (3x - 2))} = 0
\]

### Step 2: Simplify the Equation
Now simplify the expression inside the logarithm:

\[
\log_7{(x(3x - 2))} = 0
\]

This simplifies further to:

\[
\log_7{(3x^2 - 2x)} = 0
\]

### Step 3: Eliminate the Logarithm
Since \(\log_7{A} = 0\) implies \(A = 1\), we can write:

\[
3x^2 - 2x = 1
\]

### Step 4: Rearrange the Equation
Rearrange the equation to form a quadratic:

\[
3x^2 - 2x - 1 = 0
\]

### Step 5: Solve the Quadratic Equation
Now, solve the quadratic equation using the quadratic formula:

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

For \(3x^2 - 2x - 1 = 0\), the coefficients are \(a = 3\), \(b = -2\), and \(c = -1\). Substituting these into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 12}}{6}
\]

\[
x = \frac{2 \pm \sqrt{16}}{6}
\]

\[
x = \frac{2 \pm 4}{6}
\]

### Step 6: Calculate the Solutions
Now solve for the two possible values of \(x\):

\[
x = \frac{2 + 4}{6} = \frac{6}{6} = 1
\]

\[
x = \frac{2 - 4}{6} = \frac{-2}{6} = -\frac{1}{3}
\]

### Step 7: Check for Valid Solutions
Since logarithms are only defined for positive values of their arguments, we need to check which solutions are valid:

- For \(x = 1\), both \(\log_7(1)\) and \(\log_7(3(1) - 2)\) are valid.
- For \(x = -\frac{1}{3}\), the logarithmic terms become undefined (since \(x\) must be positive).

Thus, the only valid solution is:

\[
x = 1
\]

### Final Answer:
\[
x = 1
\]

Would you like more details on any of these steps, or have any further questions?

### 5 Related Questions:
1. How do you solve logarithmic equations with different bases?
2. Can you solve \(\log_2{x} + \log_2{(x - 1)} = 3\)?
3. What are the properties of logarithms used in solving equations?
4. How can you graph logarithmic functions and their transformations?
5. How do you solve logarithmic inequalities?

### Tip:
Always check the domain of the logarithmic equation to ensure solutions make the argument of the logarithms positive.

Solve the following logarithmic equation: log7x + log7(3x-2) = 0

Discover how to solve the logarithmic equation log7x + log7(3x-2) = 0 with a detailed, step-by-step solution. This guide explains how to apply logarithmic properties and solve the resulting quadratic equation using the quadratic formula. Ideal 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