Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−4)+ln(x+1)=ln(x−8)

Verified step by step guidance

1. Recall the logarithmic property that allows us to combine sums of logarithms: $\ln \left(a\right)+\ln \left(b\right)=\ln (a\cdot b)$. Use this to combine the left side: $\ln \left(x-4\right)+\ln \left(x+1\right)=\ln \left((x-4)\; (x+1)\right)$.
2. Set the equation $\ln \left((x-4)\; (x+1)\right)=\ln \left(x-8\right)$. Since the natural logarithm function is one-to-one, equate the arguments: $(x-4)\; (x+1)\; =x-8$.
3. Expand the left side: $x\cdot x+x\cdot 1\; -\; 4\cdot x-\; 4\cdot 1\; =x-\; 8$, which simplifies to $x^2+x-\; 4x-\; 4\; =x-\; 8$.
4. Combine like terms on the left: $x^2-\; 3x-\; 4\; =x-\; 8$. Then, bring all terms to one side to set the equation to zero: $x^2-\; 3x-\; 4\; -x+\; 8\; =\; 0$, which simplifies to $x^2-\; 4x+\; 4\; =\; 0$.
5. Recognize that the quadratic equation $x^2-\; 4x+\; 4\; =\; 0$ can be factored or solved using the quadratic formula. After finding the solutions, check each solution against the domain restrictions of the original logarithmic expressions: $x\; -\; 4\; >\; 0$, $x\; +\; 1\; >\; 0$, and $x\; -\; 8\; >\; 0$. Reject any solution that does not satisfy all these inequalities.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule ln(a) + ln(b) = ln(ab), is essential for combining or simplifying logarithmic expressions. This allows the equation ln(x−4) + ln(x+1) = ln(x−8) to be rewritten as a single logarithm, facilitating easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. When solving equations like ln(x−4) + ln(x+1) = ln(x−8), it is crucial to ensure that x−4 > 0, x+1 > 0, and x−8 > 0 to find valid solutions.

Solving Logarithmic Equations

After applying logarithmic properties, the equation can be converted into an algebraic form, often involving polynomials or rational expressions. Solving this algebraic equation yields potential solutions, which must then be checked against the domain restrictions to reject extraneous roots.