Solve each equation. Give solutions in exact form. See Examples 5–9. log x + log(x - 21) = log 100

Verified step by step guidance

1. Recall the logarithm property that states \( \log a + \log b = \log (a \times b) \). Use this to combine the left side of the equation: \( \log x + \log (x - 21) = \log [x(x - 21)] \).
2. Rewrite the equation using the combined logarithm: \( \log [x(x - 21)] = \log 100 \).
3. Since the logarithms are equal and the log function is one-to-one, set the arguments equal to each other: \( x(x - 21) = 100 \).
4. Expand the left side to form a quadratic equation: \( x^2 - 21x = 100 \). Then, bring all terms to one side to set the equation to zero: \( x^2 - 21x - 100 = 0 \).
5. Solve the quadratic equation \( x^2 - 21x - 100 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1 \), \( b=-21 \), and \( c=-100 \). After finding the solutions, check for extraneous solutions by ensuring \( x > 0 \) and \( x - 21 > 0 \) because the arguments of the logarithms must be positive.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the product rule, is essential. The product rule states that log(a) + log(b) = log(ab), allowing the combination of multiple logarithmic terms into a single log expression, which simplifies solving equations.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation using log properties, then converting the logarithmic form to its equivalent exponential form. This step helps isolate the variable and find exact solutions.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations like log(x) or log(x - 21), it is crucial to ensure that x and x - 21 are greater than zero to find valid solutions and exclude extraneous ones.