Solve each equation. Give solutions in exact form. See Examples 5–9. log₂ (x² - 100) - log₂ (x + 10) = 1

Verified step by step guidance

1. Recall the logarithmic property that states \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \). Apply this to combine the left side: \( \log_2 \left( \frac{x^2 - 100}{x + 10} \right) = 1 \).
2. Simplify the expression inside the logarithm. Notice that \( x^2 - 100 \) is a difference of squares and can be factored as \( (x - 10)(x + 10) \). Substitute this to get \( \log_2 \left( \frac{(x - 10)(x + 10)}{x + 10} \right) = 1 \).
3. Cancel the common factor \( x + 10 \) in the numerator and denominator, assuming \( x + 10 \neq 0 \), to simplify the logarithm to \( \log_2 (x - 10) = 1 \).
4. Rewrite the logarithmic equation in its equivalent exponential form: \( 2^1 = x - 10 \). This means \( 2 = x - 10 \).
5. Solve the resulting linear equation for \( x \) by adding 10 to both sides: \( x = 2 + 10 \). Also, check the domain restrictions to ensure the solution makes the original logarithmic expressions defined (arguments must be positive).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, especially the subtraction rule log_b(A) - log_b(B) = log_b(A/B), is essential. This allows combining or simplifying logarithmic expressions, which is crucial for solving equations involving multiple logs with the same base.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation in exponential form to isolate the variable. After simplifying the logarithmic expression, convert log_b(y) = c into y = b^c to find the exact solutions.

Domain Restrictions in Logarithms

Logarithmic functions are only defined for positive arguments. When solving equations like log_2(x^2 - 100), ensure that expressions inside the logs are greater than zero to find valid solutions and exclude extraneous roots.