To solve the equation
[tex]\[
\ln x + \ln (x - 10) = \ln (9x - 88),
\][/tex]
we can use properties of logarithms and algebraic manipulation to solve it step-by-step.
### Step-by-Step Solution
1. Combine the logarithmic terms on the left-hand side using the property [tex]\(\ln a + \ln b = \ln (ab)\)[/tex]:
[tex]\[
\ln (x(x - 10)) = \ln (9x - 88).
\][/tex]
2. Simplify the argument inside the logarithm:
[tex]\[
\ln \left(x^2 - 10x\right) = \ln (9x - 88).
\][/tex]
3. Since the natural logarithms of two expressions are equal, the expressions themselves must be equal:
[tex]\[
x^2 - 10x = 9x - 88.
\][/tex]
4. Rearrange the equation to form a standard quadratic equation:
[tex]\[
x^2 - 10x - 9x + 88 = 0 \implies x^2 - 19x + 88 = 0.
\][/tex]
5. Solve the quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -19\)[/tex], and [tex]\(c = 88\)[/tex]:
[tex]\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot 88}}{2 \cdot 1}.
\][/tex]
6. Calculate the discriminant:
[tex]\[
(-19)^2 - 4 \cdot 1 \cdot 88 = 361 - 352 = 9.
\][/tex]
7. Find the solutions:
[tex]\[
x = \frac{19 \pm \sqrt{9}}{2} = \frac{19 \pm 3}{2}.
\][/tex]
8. This gives two potential solutions:
[tex]\[
x = \frac{19 + 3}{2} = \frac{22}{2} = 11 \quad \text{and} \quad x = \frac{19 - 3}{2} = \frac{16}{2} = 8.
\][/tex]
9. Verify the solutions in the original equation to ensure they do not make the logarithmic terms undefined:
- For [tex]\(x = 11\)[/tex]:
[tex]\[
\ln 11 + \ln (11 - 10) = \ln 11 + \ln 1 = \ln 11 = \ln (9 \cdot 11 - 88) = \ln (99 - 88) = \ln 11.
\][/tex]
So, [tex]\(x = 11\)[/tex] is a valid solution.
- For [tex]\(x = 8\)[/tex]:
[tex]\[
\ln 8 + \ln (8 - 10) = \ln 8 + \ln (-2).
\][/tex]
The term [tex]\(\ln (-2)\)[/tex] is undefined in the real number system. So, [tex]\(x = 8\)[/tex] is not a valid solution.
### Conclusion
The only valid solution is [tex]\(x = 11\)[/tex]. Thus, the exact solution set is
[tex]\[
\{11\}.
\][/tex]
