Logarithmic & Exponential Functions

• Exponential functions — graphs of y = aˣ, growth vs decay, and why every one of them passes through (0,1).
• The number e — the special base where the gradient of the curve equals the curve's own height.
• Logarithmic functions — the inverse of exponentials; "the power you raise the base to, to get this number."
• ln x — the natural logarithm, base e, and how it undoes eˣ.
• Laws of logarithms — the addition, subtraction and power rules (mirrors of the laws of indices).
• Change of base — how to evaluate logs with unusual bases, with or without a calculator.
• Solving exponential equations — taking logs of both sides, including "hidden quadratic" traps.
• Linearising relationships — turning curves like y = axⁿ and y = Abˣ into straight lines using logs.

An exponential function is one where the variable sits in the power (the exponent), not the base. It has the form:

Why every graph passes through (0, 1): Whatever base you pick, anything to the power 0 is 1. So a⁰ = 1 always — meaning y = aˣ always crosses the y-axis at the point (0, 1). This is one of the most testable facts in this chapter.

The x-axis is always an asymptote. As x gets very negative (for a > 1) or very positive (for 0 < a < 1), the curve gets closer and closer to y = 0 but never actually touches it. Think of it like a rubber band stretched toward the floor — it approaches but never lands.

e ≈ 2.718...

What makes e so special? For the function y = eˣ, the gradient of the curve at any point is exactly equal to the y-value at that point. No other base has this clean property.