Know and use the function eˣ and its graph; know and use the definition of log_a x as the inverse of aˣ, where a is positive and a ≠ 1; know and use the function ln x and its graph; know and use ln x as the inverse function of eˣ

This topic introduces the exponential function eˣ and its inverse, the natural logarithm ln x. You will learn to sketch their graphs, understand their key properties, and use them to solve equations and model real-world phenomena like population growth and radioactive decay. Mastery of eˣ and ln x is essential for calculus, as they are the only functions that are their own derivatives, making them fundamental to A-Level Mathematics.

The function eˣ is defined for all real x, with a range of (0, ∞), and its graph is always increasing, crossing the y-axis at (0,1). Its inverse, ln x, is defined for x > 0, with a range of all real numbers, and its graph passes through (1,0). Understanding the relationship e^{ln x} = x and ln(e^x) = x is crucial for simplifying expressions and solving equations. You will also encounter the general logarithm log_a x as the inverse of aˣ, but the natural logarithm is the most important for calculus and further study.

In the Edexcel A-Level specification, this topic appears in both Pure Mathematics and Applications. You will use eˣ and ln x in differentiation, integration, and differential equations. A solid grasp of these functions will also help in topics like exponentials and logarithms in modelling, where you interpret growth rates and half-lives. The ability to manipulate logarithms and exponentials is a key skill assessed across multiple exam questions.