Study Notes
Exponential and logarithmic functions are interconnected, with logarithms being the inverse of exponentials. They are used to solve equations where the unknown is in the exponent.
- Exponential Function — A function of the form eˣ, where e is a constant approximately equal to 2.718. Example: e² is an exponential function.
- Natural Logarithm (ln) — The logarithm to the base e, used to find the power to which e must be raised to obtain a number. Example: ln(e) = 1.
- Laws of Logarithms — Rules that simplify expressions with logarithms, including multiplication, division, and power laws. Example: logₐx + logₐy = logₐ(xy).
- Change of Base Formula — A method to rewrite logarithms in terms of a different base. Example: logₐx = log_bx / log_ba.
Exam Tips
Key Definitions to Remember
- Exponential function: eˣ
- Natural logarithm: ln(x)
- Laws of logarithms: multiplication, division, power laws
Common Confusions
- Confusing the base of a logarithm with its argument
- Forgetting that logarithms are the inverse of exponentials
Typical Exam Questions
- What is the value of ln(e)? Answer: 1
- How do you simplify logₐx + logₐy? Answer: logₐ(xy)
- Solve the equation log₅x + 6 logₓ5 = 5. Answer: Use change of base and logarithmic laws
What Examiners Usually Test
- Understanding and applying the laws of logarithms
- Solving equations using logarithms
- Changing the base of a logarithm