Solve each equation. Give solutions in exact form. See Examples 5–9. log(3x + 5) - log(2x + 4) = 0

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule (log(a) + log(b) = log(ab)), the quotient rule (log(a) - log(b) = log(a/b)), and the power rule (n * log(a) = log(a^n)). These properties allow us to combine or simplify logarithmic expressions, which is crucial for isolating variables in equations.

Exponential Equations

Logarithmic equations can often be transformed into exponential equations. For instance, if log(a) = b, then a = 10^b (in base 10). This transformation is vital for solving equations involving logarithms, as it allows us to express the logarithmic form in a more manageable exponential form, facilitating the isolation of the variable.

Domain of Logarithmic Functions

The domain of logarithmic functions is restricted to positive arguments. For the equation log(3x + 5) - log(2x + 4) = 0 to be valid, both 3x + 5 > 0 and 2x + 4 > 0 must hold true. Understanding these restrictions is crucial for determining valid solutions and ensuring that the logarithmic expressions are defined.