Exponential & Logarithmic Functions

Exponential Functions

An exponential function has the form \(f(x) = b^x\), where \(b\) is the base and \(b > 0, b \neq 1\).

Key properties:

    \(b^0 = 1\) for any base \(b\)

    \(b^1 = b\)

    \(b^{-n} = \frac{1}{b^n}\)

    Domain: all real numbers

    Range: all positive real numbers

Logarithmic Functions

A logarithm is the inverse of an exponential function. If \(b^x = y\), then \(\log_b(y) = x\).

Common bases:

    \(\log\) (or \(\log_{10}\)): common logarithm

    \(\ln\) (or \(\log_e\)): natural logarithm

    \(\log_2\): binary logarithm

Logarithm Properties

Key properties of logarithms:

    Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)

    Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\)

    Power Rule: \(\log_b(M^n) = n \cdot \log_b(M)\)

Change of Base Formula

The change of base formula allows us to calculate logarithms with any base:

\[\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\]

This is especially useful when calculating logarithms with uncommon bases.

Practice Problems

Test your knowledge of exponential and logarithmic concepts.