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This lesson covers Exponentials and Logarithms as required by the A-Level Mathematics Pure 1 specification. Exponential functions model growth and decay in real-world contexts, while logarithms are the inverse of exponentials. Understanding the laws of logarithms and the natural logarithm ln is essential for solving a wide range of A-Level problems.
An exponential function has the form y = aˣ where a > 0 and a ≠ 1.
| Property | Description |
|---|---|
| Domain | All real numbers |
| Range | y > 0 (always positive) |
| y-intercept | (0, 1) since a⁰ = 1 |
| Asymptote | The x-axis (y = 0) is a horizontal asymptote |
| Growth/Decay | If a > 1, the function grows; if 0 < a < 1, it decays |
The number e (Euler's number) is approximately 2.71828. The function y = eˣ is the most important exponential function because its derivative is itself: d/dx(eˣ) = eˣ.
A logarithm is the inverse of an exponential. If aˣ = b, then logₐ(b) = x.
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