Solve each equation. Give solutions in exact form. ln(7 - x) + ln(1 - x) = ln (25 - x)

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Recall the logarithm property that states $ \ln a + \ln b = \ln(ab) $. Use this to combine the left side of the equation: $ \ln(7 - x) + \ln(1 - x) = \ln((7 - x)(1 - x)) $.

Rewrite the equation using the combined logarithm: $ \ln((7 - x)(1 - x)) = \ln(25 - x) $.

Since $ \ln A = \ln B $ implies $ A = B $ (assuming the domains are valid), set the arguments equal: $ (7 - x)(1 - x) = 25 - x $.

Expand the left side by multiplying the binomials: $ (7 - x)(1 - x) = 7 - 7x - x + x^2 = x^2 - 8x + 7 $.

Set up the quadratic equation by equating the expanded expression to the right side and then bring all terms to one side: $ x^2 - 8x + 7 = 25 - x $ which simplifies to $ x^2 - 7x - 18 = 0 $.

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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, especially the product rule ln(a) + ln(b) = ln(ab), is essential. This allows combining multiple logarithmic terms into a single logarithm, simplifying the equation for easier solving.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations involving ln(7 - x) and ln(1 - x), it is crucial to consider the domain restrictions 7 - x > 0 and 1 - x > 0 to ensure valid solutions.

Solving Algebraic Equations

After applying logarithmic properties, the equation reduces to an algebraic form. Solving this requires skills in manipulating and solving polynomial or rational equations, including factoring or using the quadratic formula if necessary.

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