What is e in mathematics?

e is an irrational constant, about 2.71828, defined as the limit of (1 + 1/n)ⁿ as n grows. It is the base for which the exponential function equals its own derivative.

What is the difference between log and ln?

log usually means base 10 and ln means base e. They obey the same laws, and they are linked by the change of base formula: log₁₀ x = ln x / ln 10.

Do I need to prove that (1 + 1/n)ⁿ converges?

No. The curriculum only asks you to recognise the definition; the proof that the limit exists is not required.

The Number e and the Natural Logarithm (DSE M2)

e ≈ 2.71828 is the limit of (1 + 1/n)ⁿ as n tends to infinity. The natural logarithm ln x is the logarithm to base e, so ln eˣ = x and e^{ln x} = x. e matters in M2 because eˣ is its own derivative and ln x has derivative 1/x.

Solve Introduction to e questions with AI

y = eˣ and y = ln x are mirror images in the line y = x, because each undoes the other. Try it yourself →

e and ln: Formulas

Definitione = lim_n→∞ (1 + 1/n)ⁿ

Serieseˣ = 1 + x + x²/2! + x³/3! + …

Inverse pairln eˣ = x, e^{ln x} = x (x > 0)

Any baseaˣ = e^{x ln a}

Change of baselog_a x = ln x / ln a

Log lawsln(xy) = ln x + ln y, ln(x/y) = ln x − ln y, ln xⁿ = n ln x

Worked Examples — Introduction to e

Example 1: An exponential equation that is a quadratic in eˣ

Solve e²ˣ − 3eˣ + 2 = 0.

Let u = eˣ: u² − 3u + 2 = 0
(u − 1)(u − 2) = 0
eˣ = 1 or eˣ = 2
x = 0 or x = ln 2

Answer: x = 0 or x = ln 2

Example 2: Differentiating a logarithm with another base

Find dy/dx if y = log₂ x.

Change of base: y = ln x / ln 2
1/ln 2 is a constant, so dy/dx = (1/ln 2) · (1/x)

Answer: dy/dx = 1 / (x ln 2)

💡 Exam Traps & Secrets

⚠️ ln is defined only for positive numbersIf a substitution gives eˣ = −3, reject it: eˣ is always positive. Likewise, check that every solution keeps the argument of each ln positive.

⚠️ ln(a + b) is not ln a + ln bThe log laws turn products into sums, not sums into sums. ln(x + 2) cannot be split.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Quadratic in eˣ)Solve e^x − 6e^(−x) = 1.

(Log equation)Solve ln(x + 2) + ln(x − 1) = ln 4.

(Graphs of eˣ and ln x)Show graphically that y = e^x and y = ln x are reflections of each other in y = x.