Solve for x
x
 in the following expression. (Enter your answer in exact form, i.e. not in decimal form.)
ln(7x)+7=ln(3x+2)

We are given the equation:

\[
\ln(7x) + 7 = \ln(3x + 2)
\]

### Step 1: Isolate the logarithmic expressions
Start by subtracting 7 from both sides to isolate the logarithms:

\[
\ln(7x) = \ln(3x + 2) - 7
\]

### Step 2: Eliminate the logarithms
For simpler equations, we can apply the logarithmic property:
\[
\ln(a) = \ln(b) \implies a = b
\]
So we set the arguments of the logarithms equal to each other:

\[
7x = 3x + 2
\]

### Step 3: Solve for \(x\)
Now solve the equation:

\[
7x - 3x = 2
\]

\[
4x = 2
\]

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

### Step 4: Check for extraneous solutions
Make sure that the solution satisfies the original equation, particularly that the arguments inside the logarithms are positive. For \(x = \frac{1}{2}\):

\[
7x = 7 \times \frac{1}{2} = \frac{7}{2} > 0
\]

\[
3x + 2 = 3 \times \frac{1}{2} + 2 = \frac{3}{2} + 2 = \frac{7}{2} > 0
\]

Both logarithmic arguments are positive, so \(x = \frac{1}{2}\) is valid.

Thus, the solution is:

\[
x = \frac{1}{2}
\]

Would you like further details, or have any questions?

### Related Questions:
1. How do you solve logarithmic equations with multiple terms?
2. What are the properties of logarithms used in solving equations?
3. Can logarithmic equations have no solutions?
4. How do you solve logarithmic inequalities?
5. What happens when the argument of a logarithm is negative or zero?

### Tip:
Always check for extraneous solutions when solving logarithmic equations, as the domain of logarithms requires positive arguments.

Solve for x in the following expression: ln(7x) + 7 = ln(3x + 2)

This page provides a step-by-step solution to the logarithmic equation ln(7x) + 7 = ln(3x + 2), using logarithmic properties and algebraic manipulation. Suitable for high school students, especially those in Grades 10-12.

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