AP Precalculus: Logarithms Formulas & Properties

1. Convert Between Exponential and Logarithmic Form

Exponential: \( a^x = b \)
Logarithmic: \( \log_a b = x \)
Equivalence: \( a^x = b \iff \log_a b = x \)

2. Evaluating Logarithms

\( \log_a a = 1 \)
\( \log_a 1 = 0 \)
\( \log_a a^x = x \)
\( a^{\log_a x} = x \)

Check: \(y = \log_a x \iff a^y = x \)

3. Change of Base Formula

\( \log_a b = \frac{\log_c b}{\log_c a} \)
Most common: \( \log_a b = \frac{\ln b}{\ln a} = \frac{\log_{10} b}{\log_{10} a} \)

4. Properties of Logarithms

Product: \( \log_a (xy) = \log_a x + \log_a y \)
Quotient: \( \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \)
Power: \( \log_a (x^k) = k \log_a x \)
Reverse: \( a^{x+y} = a^x\cdot a^y \), \( a^{x-y} = \frac{a^x}{a^y} \), \( (a^x)^k = a^{kx} \) to help remember properties

5. Logarithm Properties: Mixed and Evaluation

Combine: \( \log_a x^2y = 2\log_a x + \log_a y \)
Expand or condense using product, quotient, power laws as needed
Use base change and properties to compute any log (e.g., \( \log_2 8 = 3 \) because \( 2^3=8 \))