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Chapter 3: Problem 92

Use the One-to-One Property to solve the equation for \(x\). $$\ln \left(x^{2}-x\right)=\ln 6$$

Short Answer

Expert verified
The solutions for \(x\) are \(3\) and \(-2\). However, logarithms are undefined for negative numbers and zero. Substituting \(-2\) in the original equation results in \(\ln(-2)\), which is undefined. Therefore, the only solution for the equation is \(x=3\).

Step by step solution

01

Apply log properties

From the equation \(\ln \left(x^{2}-x\right)=\ln 6\), we use the One-to-one property of logarithms. If \(a = b\), then \(\ln(a) = \ln(b)\). Therefore we can simplify the equation to \(x^{2}-x=6\).
02

Rewrite the equation

Rewrite the equation as a quadratic equation, that is \(x^{2}-x-6=0\).
03

Factor the quadratic

Factor the equation as \((x-3)(x+2)=0\). This equation holds true only when either \(x-3=0\) or \(x+2=0\).
04

Solve for x

Solving \(x-3=0\) gives \(x=3\) and solving \(x+2=0\) gives \(x=-2\).

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