Study Notes
Logarithms are a way to express exponential relationships and can be used to solve equations involving powers. They are particularly useful for simplifying and evaluating expressions.
- Logarithm to Base 10 — a logarithm with base 10, often written as log(x). Example: log(100) = 2 because 10^2 = 100.
- Logarithm to Base a — a logarithm with any base 'a', written as log_a(x). Example: log_2(8) = 3 because 2^3 = 8.
- Laws of Logarithms — rules that help simplify logarithmic expressions. Example: log_b(xy) = log_b(x) + log_b(y).
Exam Tips
Key Definitions to Remember
- Logarithm to Base 10
- Logarithm to Base a
- Laws of Logarithms
Common Confusions
- Confusing the base of the logarithm with the argument
- Forgetting to apply the laws of logarithms correctly
Typical Exam Questions
- What is the value of k if log k = 2 log 3 - 5 log 2? Use the laws of logarithms to simplify and solve.
- Find the value of p if log_2 p = -1. Convert to exponential form: 2^-1 = p.
- Solve for x when 5 log 2 - log 8 = log x. Simplify using the laws of logarithms.
- Solve log x + log 3 = log 12. Use the property log_b(xy) = log_b(x) + log_b(y).
What Examiners Usually Test
- Ability to convert between exponential and logarithmic forms
- Application of the laws of logarithms to simplify expressions
- Solving equations involving logarithms