11_2.pdf

Logarithmic functions are the inverse of exponential functions, defined as \(y = \loga x\) if and only if \(a^y = x\), where \(a\) and \(x\) are positive real numbers and \(a \neq 1\). The logarithmic function with base \(a\) is denoted as \(f(x) = \loga x\). To change equations from logarithmic to exponential form, we can use the definition or the statement "log base answer = exponent".

Basic properties of logarithms include:
1. \(\loga 1 = 0\)
2. \(\loga a = 1\)
3. \(\loga a^x = x\)

Special bases for logarithms are the common log (base 10) and the natural log (base \(e\)), denoted as \(\log\) and \(\ln\), respectively. The change of base formula allows us to evaluate any log with any base: \(\loga x = \frac{\logb x}{\logb a}\) or \(\log_a x = \frac{\ln x}{\ln a}\).

Examples and exercises demonstrate how to evaluate and solve logarithmic expressions using these properties and formulas.