The number e ≈ 2.718 is the base of the exponential function eˣ, and its inverse is the natural logarithm, ln x = logₑ x. Because they are inverses, e^(ln x) = x and ln(eˣ) = x, which is how you solve equations such as e^(2x + 1) = 8 exactly. The graph of y = ln x is the reflection of y = eˣ in the line y = x.
Key facts for eˣ and ln x
- e ≈ 2.718
- e^(ln x) = x for x > 0, and ln(eˣ) = x
- ln 1 = 0 and ln e = 1
- y = eˣ: through (0, 1), asymptote y = 0
- y = ln x: through (1, 0), asymptote x = 0, defined for x > 0
The laws of logarithms and the index laws both hold for base e.
The number e and the function eˣ
e is a fixed constant, about 2.71828. The function y = eˣ is an exponential like any other, so it passes through (0, 1), is always positive and has the asymptote y = 0. The index laws still apply: e⁵/e² = e³, for example.
What makes e special is that eˣ is its own derivative, so it is the natural base for continuous growth and decay and appears throughout calculus.
ln x as the inverse of eˣ
The natural logarithm ln x is the logarithm to base e, so it is the inverse of eˣ: each undoes the other. That gives two key results, e^(ln x) = x and ln(eˣ) = x, and two values worth memorising, ln 1 = 0 and ln e = 1. As with any logarithm, you can only take ln of a positive number.
Transformations of eˣ and ln x
For y = e^(ax + b) + c with a > 0, the asymptote is y = c and the y-intercept is e^b + c. For y = ln(ax + b), the argument must be positive, so there is a vertical asymptote where ax + b = 0. A translation changes the asymptote; a stretch from the y-axis does not move a horizontal asymptote.
Solving exponential and log equations exactly
Undo each function with its inverse. e^(ax + b) = p gives ax + b = ln p, and ln(ax + b) = q gives ax + b = e^q. Then solve the linear equation, leaving the answer in terms of ln or e if an exact answer is asked for. Because eˣ is always positive, an equation like eˣ = −4 has no real solution.
Quadratics in eˣ and inverse functions
Since e²ˣ = (eˣ)², an equation such as e²ˣ − 5eˣ + 6 = 0 is a quadratic in disguise. Substitute u = eˣ, solve for u, then take ln of each positive root; reject any root that is zero or negative. To find the inverse of a function built from eˣ, write y = f(x) and make x the subject using ln.
Worked examples
Example 1. Simplify e⁵/e², and state the y-intercept and asymptote of y = eˣ.
- e⁵/e² = e⁵⁻² = e³.
- y = eˣ passes through (0, 1) and has asymptote y = 0.
Answer: e³; intercept (0, 1); asymptote y = 0.
Marks: A1 for e³, B1 for the intercept and asymptote.
Example 2. Evaluate ln e³, ln 1 and e^(ln 5).
- ln undoes e, so ln e³ = 3.
- e⁰ = 1, so ln 1 = 0.
- e undoes ln, so e^(ln 5) = 5.
Answer: 3, 0 and 5.
Marks: B1 for each value.
Example 3. State the asymptote and y-intercept of y = e³ˣ − 2.
- As x → −∞, e³ˣ → 0, so the asymptote is y = −2.
- At x = 0: y = e⁰ − 2 = −1.
Answer: Asymptote y = −2, intercept (0, −1).
Marks: B1 for the asymptote, B1 for the intercept.
Example 4. Solve e^(2x + 1) = 8, giving your answer exactly.
- Take ln of both sides: 2x + 1 = ln 8.
- 2x = ln 8 − 1.
Answer: x = (ln 8 − 1)/2.
Marks: M1 for 2x + 1 = ln 8, A1 for the exact answer.
Example 5. Solve e²ˣ − 5eˣ + 6 = 0.
- Let u = eˣ: u² − 5u + 6 = 0.
- (u − 2)(u − 3) = 0, so u = 2 or u = 3.
- eˣ = 2 gives x = ln 2; eˣ = 3 gives x = ln 3.
Answer: x = ln 2 or x = ln 3.
Marks: M1 for the quadratic in u, A1 for u = 2, 3, A1 for both values of x.
Common mistakes
- Taking ln of a negative number. eˣ = −4 has no real solution; ln is only defined for positive numbers.
- Writing ln(a + b) = ln a + ln b. There is no law for the log of a sum.
- Not isolating the exponential first. For 3e²ˣ = 12, divide by 3 before taking ln.
- Rounding when an exact answer is asked. Leave answers as ln 8 or (ln 8 − 1)/2 if the question says exact.
- Keeping a negative root in a quadratic in eˣ. eˣ is always positive, so reject u ≤ 0.
How eˣ and ln x are examined
In Edexcel International A-Level, eˣ and ln x are part of Pure 3 (WMA13), in the exponentials and logarithms chapter. In Edexcel A-Level (9MA0) they are first met in Year 1. Questions typically ask you to:
- state the asymptote and intercept of a transformed exponential or log graph
- solve an equation exactly, leaving ln or e in the answer
- solve a quadratic in eˣ, rejecting impossible roots
- find the inverse of a function involving eˣ or ln x
Exam-style questions, with full mark schemes, are in the Pure 3 exponentials and logarithms exam questions. The Pure 3 exponentials and logarithms worksheet and the revision engine for logarithms give further practice.
This builds on introducing logarithms. Next, exponential growth and decay uses eˣ to model real situations.
Frequently asked questions
What is e?
An irrational constant, about 2.718. It is the base for which the exponential function is its own derivative.
What is the difference between ln and log?
ln is the logarithm to base e. log usually means base 10.
How do you solve eˣ = 5?
Take ln of both sides: x = ln 5, which is about 1.61.