Maths › Exponentials and logarithms › Exponential functions and e
Exponential functions and e
Put x in the exponent and everything changes. Equal steps in x now multiply y instead of adding to it. Among all the possible bases one is special, because the curve of e to the x climbs at a rate equal to its own height, and that single property is why the same number keeps turning up in banking, biology and radioactive decay.
Builds on Indices and surds and Graphs, proportion and transformations.
IN THIS TOPIC
- Sketch y = ax for any positive base except 1, growth and decay alike, with the right asymptote.
- Use the gradient property of ekx, and transform e-curves into the form eax+b + c.
- Recognise when a situation calls for an exponential model, and read its constants.
COMMON MISCONCEPTION
Exponential growth means growing very fast.
The family a to the x
The function y = ax, for positive a other than 1, puts the variable up in the exponent. Every unit step in x now multiplies y by a instead of adding anything to it. Every member of the family passes through (0, 1), because a0 = 1 straight from the index laws, and the x-axis is a horizontal asymptote on one side. Which side depends on the base. With a > 1 the curve grows to the right; with 0 < a < 1 it decays, and the two shapes are mirror images because (1/2)x = 2−x. Base 1 is left out for a reason. 1x = 1 for every x, so it draws the flat line y = 1, which grows nowhere, decays nowhere, and has no asymptote to approach.
Every exponential is positive everywhere. The curve hugs its asymptote and never touches it, so ax = 0 has no solution at all, and exam questions poke at that fact from several directions.
The special base e
Compare gradients at the point every one of these curves shares, (0, 1). Base 2 leaves it with gradient about 0.69 and base 3 with about 1.10, so somewhere between the two sits a base that leaves with gradient 1 precisely. That base is e = 2.71828…, and the property does not stop at the one point. It holds along the whole curve, and for any multiple of x,
so the gradient of ekx anywhere is k times the height there.
WORKED EXAMPLE
A gradient with no calculus in sight
Find the gradient of y = e2x at the point where x = 1.
The gradient function is 2e2x, straight from the boxed property with k = 2.
At x = 1 that gives 2e2 = 14.8 to 3 significant figures.
The height there is e2 ≈ 7.39, and the gradient came out at twice the height. That is what k = 2 promises at every point on this curve.
GUIDED PRACTICE
A transformed exponential
Sketch y = e−x − 2, stating the equation of its asymptote and its y-intercept, before opening the working.
Show the working
Build it in two moves from y = ex. The −x reflects in the y-axis and turns growth into decay, then the −2 translates the whole thing down 2.
The asymptote travels with the curve, down to y = −2. The y-intercept is e0 − 2 = −1.
So the curve falls from the top left, cuts the y-axis at −1, and flattens onto y = −2 without ever reaching it. The transformations lesson runs this whole show, and nothing about e changes its rules.
Why nature keeps choosing e
The deep reason exponentials model so much is the gradient property read backwards. Suppose a quantity's rate of change is proportional to how much of it there is. More bacteria breed more bacteria. More atoms decay more often. More money earns more interest. Any quantity behaving that way has to follow P = Aekt.
Growth of this kind can start extremely slowly. The defining feature is the proportionality, and a small k makes for a very leisurely curve indeed.
INDEPENDENT PRACTICE
Reading a growth model
A population is modelled by P = 500e0.2t, with t in days. Write down the initial population, find P after 5 days, and explain what 0.2 says about the growth.
Show the working
At t = 0 the exponential is 1, so the initial population is 500.
After 5 days, P = 500e1 = 500e ≈ 1360 to 3 significant figures.
The 0.2 is the proportionality constant. At every moment the population grows at 0.2 × its current size per day. Early on that is only 100 a day. By day 5 it is about 272 a day, out of the very same rule.
A question asking for the “initial” value always means t = 0, and the answer is the constant out in front whenever the model is written as Aekt.
ASSESSMENT FOCUS
- Every ax sketch needs three things labelled. The point (0, 1), the asymptote, and the correct direction of growth or decay for the base you were given.
- Quote the gradient property before you use it. Writing “gradient of ekx is kekx” is itself a mark.
- Transformations of e-curves obey the standard rules, and the asymptote moves with the curve. State its new equation in full.
- In P = Aekt, A is the value at t = 0 and k is the per-unit-time proportionality constant. Interpret both with their units attached.
- ax is never zero and never negative. “Explain why the model predicts P never reaches 0” is asking you to talk about the asymptote.
CHECK YOURSELF
For the curve y = e3x, write down the gradient function, the gradient at x = 0, and the y-intercept.
Show a hint
One boxed property answers all three parts.
Show the answer
The gradient function is 3e3x, by the property with k = 3.
At x = 0 the gradient is 3e0 = 3, three times the height there.
The y-intercept is e0 = 1, shared by the whole exponential family.
Every base's curve passes through (0, 1) and hugs an asymptote it never touches.
e is the base whose gradient equals its height; e to the kx grows at k times itself.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the exponential functions and e questions page.
CHECK YOUR PROGRESS
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- Sketch y = ax for any positive base except 1, growth and decay alike, with the right asymptote.
- Use the gradient property of ekx, and transform e-curves into the form eax+b + c.
- Recognise when a situation calls for an exponential model, and read its constants.
Open the full revision checklist to see every objective in the course in one place.