Exponential_and_Logarithmic_Functions.pptx

Exponential_and_Logarithmic_Functions

1. Exponential & Logarithmic Functions Understanding $e^x$ and $\ln x$

2. Warm-Up: Indices Recall the laws of indices: - a^m * a^n = a^(m+n) - (a^m)^n = a^(mn) Example: Simplify a^3 * a^2

3. The Exponential Function y = e^x Properties: - Passes through (0,1) - Always positive - Increases as x increases - As x→-∞, y→0

4. Why e? e is a special base: - Derivative of e^x is itself - Appears naturally in growth/decay problems

5. Introducing ln(x) Definition: - ln(x) = log_e(x) - Means 'the power you put on e to get x' Examples: - ln(1)=0 - ln(e)=1

6. Properties of ln(x) Laws: - ln(ab) = ln a + ln b - ln(a/b) = ln a - ln b - ln(a^k) = k ln a

7. Graphs of e^x and ln(x) Graphs: - y=e^x passes (0,1), increases rapidly - y=ln(x) passes (1,0), only defined for x>0 - They are reflections across y=x

8. Key Identities Inverse Relationships: - e^(ln x) = x (x>0) - ln(e^x) = x (all real x)

9. Practice Questions 1. Simplify: e^(ln 9) 2. Solve: e^(2x) = 10 3. Solve: ln(5x) = 3

10. Summary • Exponential and logarithm are inverse functions • Graphs are reflections in y=x • Laws of logs essential for solving equations