Solve the equation. Give an exact solution.
ln(3x - 2) = ln 2 - ln(x - 1)
ln(3x - 2) = ln 2 - ln(x - 1)
ln(3x-2) = ln(2/(x-1))
3x-2 = 2/(x-1)
3x^2 - 5x + 2 = 2
x(3x-5) = 0
x = 0 or x = 5/3
BUT, if x=0 , ln(x-1) would be undefined,
so x = 5/3
To solve the equation ln(3x - 2) = ln 2 - ln(x - 1), we can use the properties of logarithms.
First, we can simplify the right side of the equation by applying the quotient property of logarithms, which states that ln(a) - ln(b) = ln(a / b). Applying this property, we have:
ln(3x - 2) = ln(2 / (x - 1))
Now that both sides of the equation have the same base (ln), we can equate the expressions inside the logarithm. Therefore, we have:
3x - 2 = 2 / (x - 1)
Next, we can solve for x by cross-multiplying:
(3x - 2)(x - 1) = 2
Expanding the left side of the equation:
3x^2 - 3x - 2x + 2 = 2
Simplifying:
3x^2 - 5x = 0
Factorizing the equation:
x(3x - 5) = 0
Setting each factor to zero and solving for x:
x = 0 or 3x - 5 = 0
For x = 0, we need to check if it is a valid solution by substituting it back into the original equation. However, ln(3(0) - 2) is undefined, which means x = 0 is not a valid solution.
For 3x - 5 = 0, we can solve for x:
3x = 5
x = 5/3
Therefore, the exact solution to the equation ln(3x - 2) = ln 2 - ln(x - 1) is x = 5/3.