Summary
Logarithmic and exponential functions involve understanding the relationship between logarithms and indices, using laws of logarithms, and solving equations and inequalities. These functions also include transforming relationships to linear form using logarithms.
- Logarithms to base 10 — The inverse of raising 10 to a power. Example: log₁₀1000 = 3 because 10³ = 1000.
- Logarithms to base a — The inverse of raising a to a power. Example: log₄64 = 3 because 4³ = 64.
- Laws of logarithms — Rules for simplifying logarithmic expressions. Example: logₐx + logₐy = logₐ(xy).
- Natural logarithms — Logarithms with base e, denoted as ln. Example: ln(e²) = 2.
- Exponential functions — Functions involving powers of a constant base. Example: eˣ is the natural exponential function.
Exam Tips
Key Definitions to Remember
- Logarithm: The power to which a base must be raised to produce a given number.
- Exponential function: A function where the variable is in the exponent.
- Natural logarithm: A logarithm with base e.
Common Confusions
- Confusing the base of a logarithm with its argument.
- Forgetting to reverse inequality signs when solving exponential inequalities with bases between 0 and 1.
Typical Exam Questions
- Solve log₁₀x = 2. What is x? x = 100
- Simplify log₄16 + log₄4. log₄64 = 3
- Solve the equation eˣ = 5. x = ln(5)
What Examiners Usually Test
- Application of logarithmic laws to simplify expressions.
- Solving logarithmic and exponential equations.
- Transforming non-linear relationships to linear form using logarithms.