Question

solve for x.
ln(x) + ln(x - 2) = ln(3)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution is x =
(type an exact answer in simplified form. use a comma to separate answers as needed.)
b. there is no solution.

Explanation:

Step1: Apply Logarithm Property

Using the property \(\ln a + \ln b=\ln(ab)\), we rewrite the left - hand side of the equation \(\ln(x)+\ln(x - 2)=\ln(3)\) as \(\ln(x(x - 2))=\ln(3)\).

Step2: Use One - to - One Property of Logarithms

Since the natural logarithm function \(y = \ln u\) is one - to - one (if \(\ln u=\ln v\), then \(u = v\) for \(u>0\) and \(v>0\)), we can set the arguments equal to each other: \(x(x - 2)=3\).

Step3: Expand and Solve the Quadratic Equation

Expand the left - hand side: \(x^{2}-2x = 3\).
Rearrange it to the standard quadratic form \(ax^{2}+bx + c = 0\): \(x^{2}-2x-3 = 0\).
Factor the quadratic equation: We need two numbers that multiply to \(-3\) and add up to \(-2\). The numbers are \(-3\) and \(1\). So, \(x^{2}-2x - 3=(x - 3)(x+1)=0\).
Set each factor equal to zero:

• If \(x - 3=0\), then \(x = 3\).
• If \(x + 1=0\), then \(x=-1\).

Step4: Check the Domain of the Logarithm

The domain of \(y=\ln x\) is \(x>0\).

• For \(x = 3\): \(\ln(3)\) is defined (since \(3>0\)) and \(\ln(3 - 2)=\ln(1)=0\) is also defined.
• For \(x=-1\): \(\ln(-1)\) is not defined (because the argument of a natural logarithm must be positive). So we reject \(x=-1\).

Answer:

A. The solution is \(x = 3\)