Exponential and logarithmic functions (IB Math AA SL)

Key ideas

• eˣ and ln x are inverse functions, so their graphs are reflections in y = x.
• eˣ > 0 for every x: its range is y > 0 and y = 0 is a horizontal asymptote. ln x is defined only for x > 0, with the vertical asymptote x = 0.
• ln 1 = 0 and ln e = 1. Laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aⁿ) = n ln a.
• To solve aˣ = b, take logs of both sides: x = ln b / ln a.
• An equation in e²ˣ and eˣ is a quadratic in disguise: put u = eˣ, solve for u, then reject any u ≤ 0, since eˣ is always positive.
• Derivatives: the derivative of eˣ is eˣ and the derivative of ln x is 1/x.

Worked example

Solve e²ˣ − eˣ − 6 = 0.

1. Let u = eˣ, so e²ˣ = u², and the equation is u² − u − 6 = 0.
2. Factorise: (u − 3)(u + 2) = 0, so u = 3 or u = −2.
3. eˣ = −2 has no solution, because eˣ > 0.
4. eˣ = 3 gives x = ln 3.

Answer: x = ln 3 (the only solution).

Exam-style SL AA questions on this topic

• Exponential and Logarithmic Functions questions — 25 questions with mark schemes, plus free worked examples
• All SL AA Functions questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Tables of values on your calculator

See it on a graph

• Graph of y = eˣ
• Graph of y = ln x
• Open the free Grapher

Maths IA ideas that use exponential and logarithmic functions

• Radioactive decay with dice: does chance produce an exponential curve?
• How long would it take to guess your password? Counting and logarithms
• Exponential or logistic? Modelling my country's population
• How quickly does a swing die down? Exponential or linear decay?
• All SL AA IA ideas

Related in Functions

• Rational functions and asymptotes
• Domain, range, and function composition
• Function transformations — translations, stretches, reflections

Revise Exponential and logarithmic functions for the exam

• SL AA revision notes — every unit summarised, with the key results to learn
• SL AA past papers — specimen papers and timed practice papers
• Formulas for Functions — on the one-page SL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: How do the assumptions built into a model shape the conclusions it can reach?

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S87 · Problem S22 · Problem S86 · Problem S93. All 22 →