Exponential & Logarithms

At a Glance

1. What is e?

The Definition

e is an irrational number — like π, it cannot be written as a simple fraction. Its value is approximately 2.718. But e isn't just a random constant. It's the base of the most important exponential function in mathematics and science.

Think of it this way: if you have a bank account earning 100% interest compounded continuously, e is the limit of how much your money grows. No matter what base you pick for exponential growth, e shows up naturally in calculus and physics.

The Special Property of e

Here's what makes e truly special:

In other words: the gradient of y = e^x at any point is equal to the value of the function at that point. If you zoom in at x = 3, the curve is climbing at rate e³. This self-replicating property is why e appears everywhere in real-world modelling.

Sketching y = e^x

The curve y = e^x has these features:

• y-intercept: passes through (0, 1) — because e⁰ = 1
• Asymptote: the x-axis (y = 0) as x → −∞; the curve never touches it
• Shape: increases very slowly at first, then explosively fast as x grows

Comparing Exponential Bases

How does y = e^x compare to y = 2^x or y = 3^x? All three pass through (0, 1), but they grow at different rates. Since e ≈ 2.718, the curve y = e^x grows faster than y = 2^x but slower than y = 3^x. More importantly, only y = e^x has the special derivative property above.

The Reciprocal: y = e^(−x)

The graph y = e^(−x) is a reflection of y = e^x in the y-axis. Instead of growing, it decays towards zero:

• y-intercept: (0, 1)
• Asymptote: y = 0 as x → ∞
• Shape: decreases rapidly at first, then flattens out

2. Exponential Growth & Decay

3. Natural Logarithm (ln)

The Definition

ln stands for natural logarithm. It is a function (not a number), and it is defined as the logarithm with base e: