Question

In the following exercises, solve for x.

$\ln \left(3x-2\right)=\ln \left(x+4\right)+\ln 2$

Short answer

The value of x for the equation$\ln \left(3x-2\right)=\ln \left(x+4\right)+\ln 2$ is$10$

Step-by-step solution

Step 1. Given data 

The given Logarithmic Equation is$\ln \left(3x-2\right)=\ln \left(x+4\right)+\ln 2$

Step 2. Apply product property of the logarithm

Use product property of the logarithm

$$\begin{gathered} \ln \left(3x-2\right)=\ln \left(x+4\right)+\ln 2 \\ \ln \left(3x-2\right)=\ln \left(\left(x+4\right)2\right) \\ \ln \left(3x-2\right)=\ln \left(2x+8\right) \end{gathered}$$

Step 3. Apply one to one property of the logarithm

Use one to one property of the logarithm

$$\begin{gathered} \ln \left(3x-2\right)=\ln \left(2x+8\right) \\ 3x-2=2x+8 \\ 3x-2x=8+2 \\ x=10 \end{gathered}$$

Step 4. verification

Substitute $10$ for x in the equation

$$\begin{gathered} \ln \left(3x-2\right)=\ln \left(2x+8\right) \\ \ln \left(3\left(10\right)-2\right)\stackrel{?}{=}\ln \left(2\left(10\right)+8\right) \\ \ln \left(30-2\right)\stackrel{?}{=}\ln \left(20+8\right) \\ \ln \left(28\right)=\ln \left(28\right) \end{gathered}$$