Exponential Growth and Decay

Exponential functions look like \( f(x) = a \cdot b^x \), where the variable is in the exponent. They're perfect for modeling population growth, radioactive decay, and interest in bank accounts.

Key Features

• Growth: When \( b > 1 \), the function increases rapidly.
• Decay: When \( 0 < b < 1 \), the function decreases quickly.

Logarithms: The Inverse Heroes

Logarithms answer the question: "To what exponent must we raise the base to get this number?" If \( b^y = x \), then \( \log_b(x) = y \).

Properties

• Logarithm of a Product: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Change of Base: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \)

Where They Show Up

Exponential and logarithmic functions are used in science for measuring sound intensity (decibels), earthquake strength (Richter scale), and in finance for compound interest.