To solve the equation [tex]\(\ln 2 + \ln (x - 4) = \ln x\)[/tex], we will use properties of logarithms and basic algebraic manipulation. Here's the step-by-step solution:
1. Combine the logarithms on the left-hand side using the property of logarithms [tex]\(\ln a + \ln b = \ln (a \cdot b)\)[/tex]:
[tex]\[
\ln 2 + \ln (x - 4) = \ln (2 \cdot (x - 4))
\][/tex]
Therefore, the equation becomes:
[tex]\[
\ln (2(x - 4)) = \ln x
\][/tex]
2. Since the natural logarithm ([tex]\(\ln\)[/tex]) function is one-to-one, the arguments of the logarithms must be equal if the logarithms are equal:
[tex]\[
2(x - 4) = x
\][/tex]
3. Simplify the equation by distributing and solving for [tex]\(x\)[/tex]:
Distribute 2:
[tex]\[
2x - 8 = x
\][/tex]
4. Isolate [tex]\(x\)[/tex] by moving all [tex]\(x\)[/tex]-terms to one side of the equation:
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
2x - x - 8 = 0
\][/tex]
Simplify:
[tex]\[
x - 8 = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Add 8 to both sides:
[tex]\[
x = 8
\][/tex]
Thus, the solution to the equation [tex]\(\ln 2 + \ln (x - 4) = \ln x\)[/tex] is:
[tex]\[
x = 8
\][/tex]
