Solve: log₂ (x+9) — log₂ x = 1.

Verified step by step guidance

1. Recall the logarithmic property that states: \(\log_a b - \log_a c = \log_a \left( \frac{b}{c} \right)\), which allows us to combine the difference of two logs with the same base into a single log of a quotient.
2. Apply this property to the equation \(\log_2 (x+9) - \log_2 x = 1\) to rewrite it as \(\log_2 \left( \frac{x+9}{x} \right) = 1\).
3. Rewrite the logarithmic equation in its equivalent exponential form: \(\frac{x+9}{x} = 2^1\).
4. Simplify the right side to get \(\frac{x+9}{x} = 2\), then multiply both sides by \(x\) to clear the denominator, resulting in \(x + 9 = 2x\).
5. Solve the linear equation \(x + 9 = 2x\) by isolating \(x\), and then check the solution to ensure it does not make any logarithm argument negative or zero.

Key Concepts

Properties of Logarithms

Logarithmic properties, such as the subtraction rule log_b(A) - log_b(B) = log_b(A/B), allow simplification of expressions involving logs. This property is essential for combining or breaking down logarithmic terms to solve equations.

Definition of Logarithms

A logarithm log_b(A) answers the question: to what power must the base b be raised to get A? Understanding this helps convert logarithmic equations into exponential form, which is often easier to solve.

Domain Restrictions in Logarithmic Functions

The argument of a logarithm must be positive, so expressions inside logs (like x and x+9) must be greater than zero. Recognizing domain restrictions ensures solutions are valid within the function's domain.