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
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
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: \(\log_b(M^n) = n \cdot \log_b(M)\)
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.
Test your knowledge of exponential and logarithmic concepts.