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)

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Recall the logarithmic property that allows us to combine sums of logarithms: \(\ln a + \ln b = \ln(ab)\). Apply this to the left side of the equation to combine the logarithms: \(\ln(x - 4) + \ln(x + 1) = \ln((x - 4)(x + 1))\).

Rewrite the equation using the combined logarithm: \(\ln((x - 4)(x + 1)) = \ln(x - 8)\).

Since the natural logarithm function \(\ln\) is one-to-one, set the arguments equal to each other: \((x - 4)(x + 1) = x - 8\).

Expand the left side: \(x^2 + x - 4x - 4 = x - 8\), which simplifies to \(x^2 - 3x - 4 = x - 8\).

Bring all terms to one side to form a quadratic equation: \(x^2 - 3x - 4 - x + 8 = 0\), which simplifies to \(x^2 - 4x + 4 = 0\). Then solve this quadratic equation for \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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), ensure that x−4 > 0, x+1 > 0, and x−8 > 0 to find valid solutions and reject extraneous roots.

Solving Logarithmic Equations

After applying logarithmic properties, convert the equation from logarithmic form to an algebraic equation to solve for x. For example, if ln(A) = ln(B), then A = B. This step simplifies the problem to solving polynomial or rational equations, which can then be checked against the domain restrictions.

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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−2)−ln(x+3)=ln(x−1)−ln(x+7)

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