Overview

This topic focuses on exponential and logarithmic functions, their properties, graphs, and how to solve equations involving them. You’ll explore the relationship between logarithms and exponents, and learn techniques for solving growth, decay, and compound interest problems.

Key Concepts and Structures

• Exponential Functions: Have the form f(x) = a^x where a > 0 and a ≠ 1. Graphs increase or decrease rapidly.
• Logarithmic Functions: Inverse of exponential functions. log_b(x) means the exponent you raise b to in order to get x.
• Basic Rules:
  • log_b(b^x) = x
  • b^{log_b(x)} = x
  • 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)
• Natural Logarithm: ln(x) is log base e (Euler’s number ≈ 2.718).
• Change of Base Formula: log_b(x) = log_c(x) / log_c(b) (commonly use base 10 or base e)
• Solving Exponential Equations: Isolate base, then take log of both sides.
• Solving Logarithmic Equations: Combine logs, rewrite in exponential form, and solve.
• Applications: Compound interest, population growth, radioactive decay.

Step-by-Step Example

Problem: Solve 2^x = 16

Step 1: Rewrite 16 as a power of 2:
16 = 2^4

Step 2: Set exponents equal:
2^x = 2^4 → x = 4

Final Answer: x = 4

Quick Tip