Overview

This topic focuses on exponential and logarithmic functions, their properties, graphs, and applications. These functions are used to model growth, decay, and real-world situations involving repeated multiplication or inverse relationships.

Key Concepts and Properties

• Exponential Function: Has the form f(x) = a^x, where a > 0 and a \ne 1.
• Logarithmic Function: Inverse of exponential functions. y = \log_b(x) is equivalent to b^y = x.
• Properties of Logs:
  • \log_b(xy) = \log_b(x) + \log_b(y)
  • \log_b(x/y) = \log_b(x) - \log_b(y)
  • \log_b(x^r) = r\log_b(x)
• Common and Natural Logs: Common log: \log(x), base 10; Natural log: \ln(x), base e.
• Graphing: Exponential graphs show rapid growth/decay; logarithmic graphs increase slowly and pass through (1, 0).
• Domain and Range: Exponential: Domain = all real numbers, Range = (0, \infty); Logarithmic: Domain = (0, \infty), Range = all real numbers.

Step-by-Step Example

Problem: Solve for x: 2^{x + 1} = 16

Step 1: Rewrite 16 as a power of 2:
2^{x + 1} = 2^4

Step 2: Since the bases are equal, set the exponents equal:
x + 1 = 4

Step 3: Solve for x:
x = 3

Answer: x = 3

Quick Tip