Question Find the solution to the equation $\ln (x-1)=1+\ln (3x+2)$ .

Studdy Solution

STEP 1

Assumptions
1. The equation to solve is $\ln (x-1) = 1 + \ln (3x+2)$ .
2. We assume that $x$ is in the domain of both logarithmic functions, meaning $x>1$ and $3x+2>0$ .
3. We will use properties of logarithms to solve the equation.

STEP 2

First, we need to simplify the equation using logarithmic properties. We can combine the logarithms on the right-hand side using the property $\ln a + \ln b = \ln(ab)$ .
$\ln (x-1) = \ln(e) + \ln (3x+2)$

STEP 3

Since $\ln(e) = 1$ , we can rewrite the equation as:
$\ln (x-1) = \ln(e \cdot (3x+2))$

STEP 4

Now, apply the property $\ln a = \ln b \Rightarrow a = b$ to equate the arguments of the logarithms.
$(x-1) = e \cdot (3x+2)$

STEP 5

Distribute $e$ on the right-hand side of the equation.
$(x-1) = 3ex + 2e$

STEP 6

Now, we need to isolate $x$ . First, move all terms involving $x$ to one side and constants to the other side.
$x - 3ex = 2e + 1$

STEP 7

Factor out $x$ on the left-hand side.
$x(1 - 3e) = 2e + 1$

STEP 8

Divide both sides by $(1 - 3e)$ to solve for $x$ .
$x = \frac{2e + 1}{1 - 3e}$

STEP 9

We need to simplify the right-hand side. Since there is no simple form for this expression, we can check the given options to see if any of them match the simplified form of $x$ .

STEP 10

We can see that none of the given options directly match the form of $x$ we have derived. Therefore, we can conclude that the correct answer is "none".
The most correct answer for the solutions of the equation $\ln (x-1) = 1 + \ln (3x+2)$ is "none".

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms