Condense right side. To do so, apply the product property of logarithm which is `ln M + ln N = ln (M*N)` .
`ln(3x-1)=ln(2(x-2)) + ln(x+1)`
Since both side have same base (natural logarithm), equate their argument equal to each other.
`3x - 1= 2 (x - 2)(x + 1)`
Then, expand right side. So FOIL (x-2)(x+1).
`3x - 1 = 2(x^2-x-2)`
Then, distribute 2 to x^2-x- 2.
`3x-1 = 2x^2-2x-4`
Then, set one side equal to zero by subtract 3x and adding 1 to both sides of the equation.
Next, factor right side.
Then, set each factor equal to zero and solve for x.
For the first factor,
And for the second factor,
Since in logarithm a negative argument is not allowed, x=-1/2 is not considered as a solution.
Hence, the solution to the given logarithmic equation is `x=3` .