IB Maths AA HLTopic 2 — FunctionsPaper 1 & 2~8 min read
Exponential & Logarithmic Functions
Exponentials and logarithms are inverse functions — that one fact does most of the work in this section. Exponentials grow (or decay) by repeatedly multiplying. Logs undo them. Knowing the standard graphs, the key features, and how the two are linked sets you up for everything from compound interest to half-life problems to differential equations later on.
📘 What you need to know
Exponential function: f(x) = ax, a > 0. Domain: all reals; range: f(x) > 0.
Logarithmic function: f(x) = logax, x > 0. Domain: x > 0; range: all reals.
Inverses: logax and ax undo each other. So loga(ax) = x and alogax = x.
Natural ones: ex and ln x are the same functions with base e ≈ 2.718. They’re the most useful for calculus later.
Convert any base to e: ax = ex ln a. In the formula booklet.
Exponential graph features: y-intercept (0, 1); horizontal asymptote y = 0; passes through (1, a); no roots; no max/min.
Logarithmic graph features: x-intercept (1, 0); vertical asymptote x = 0; passes through (a, 1); no y-intercept; no max/min.
Reflection: y = logax is the reflection of y = ax in the line y = x.
The exponential function — y = ax
Exponential functionf(x) = ax, a > 0
domain: x ∈ ℝ · range: f(x) > 0
Key features
y-intercept
(0, 1)
always — because a0 = 1
Anchor point
(1, a)
tells you the base from the graph
Horizontal asymptote
y = 0
graph hugs but never touches the x-axis
y = aˣ — two cases depending on a
If a > 1, the function grows ever faster as x increases. If 0 < a < 1, the function decays toward zero as x increases. Either way, the curve never crosses the x-axis — output is always positive.
The natural exponential
The most-used exponential is y = ex, where e ≈ 2.718 is Euler’s number. It’s the one that pops up in calculus (its own derivative), in compound interest (continuously compounded), and almost any natural growth/decay process.
Convert any base to eax = ex ln a✓ in formula booklet
The logarithmic function — y = logax
Logarithmic functionf(x) = logax, x > 0
domain: x > 0 · range: f(x) ∈ ℝ
Key features
x-intercept
(1, 0)
always — because loga 1 = 0
Anchor point
(a, 1)
tells you the base from the graph
Vertical asymptote
x = 0
graph never touches the y-axis
The natural logarithm
Just as ex is the natural exponential, ln x = logex is the natural logarithm. They’re inverses of each other:
Inverse property of e and ln
ln(ex) = x and eln x = x
Change of base
logax = ln xln a✓ in formula booklet
The inverse relationship — reflections in y = x
Since logs and exponentials are inverses of each other, their graphs are mirror images across the line y = x. Flipping all the (x, y) pairs swaps the key features:
Exponential y = ax
(0, 1) and (1, a)
y-intercept (0, 1); horizontal asymptote y = 0
Logarithmic y = logax
(1, 0) and (a, 1)
x-intercept (1, 0); vertical asymptote x = 0
y = eˣ and y = ln x — reflections in y = x
🤔 Why does this matter?
Whenever you have an exponential equation you can’t solve algebraically directly (e.g. ex = 7), apply the inverse — take ln of both sides — and the unknown comes loose: x = ln 7. That trick alone solves a huge slice of exam problems.
Worked examples
WE 1
Identify features of an exponential function
For f(x) = 5x, state: (a) the y-intercept, (b) the horizontal asymptote, (c) the value of f(2) and f(−1).
(a) y-intercept: substitute x = 0f(0) = 5⁰ = 1 → (0, 1)(b) Horizontal asymptote: as x → −∞, 5ˣ → 0y = 0(c) Substitute x = 2 and x = −1f(2) = 25, f(−1) = 1/5(a) (0, 1); (b) y = 0; (c) f(2) = 25, f(−1) = 1/5negative exponents give reciprocals: 5⁻¹ = 1/5
WE 2
Rewrite an exponential using base e
Express y = 7x in the form y = ekx for some constant k.
Use the identity aˣ = eˣ ln a7ˣ = eˣ ln 7y = e^(x ln 7), so k = ln 7this trick is essential for calculus — derivatives of bases other than e require this conversion
WE 3
Identify features of a logarithmic function
For f(x) = log4x, state: (a) the x-intercept, (b) the vertical asymptote, (c) the value of f(16) and f(1/4).
(a) x-intercept: solve log₄ x = 0x = 4⁰ = 1 → (1, 0)(b) Vertical asymptote: log only defined for x > 0x = 0(c) Use definition: log₄ y = z ⟺ 4ᶻ = yf(16) = log₄ 16 = 2 (since 4² = 16)f(1/4) = log₄(1/4) = −1 (since 4⁻¹ = 1/4)(a) (1, 0); (b) x = 0; (c) f(16) = 2, f(1/4) = −1log of a number less than 1 (but positive) is always negative
WE 4
Find the inverse of a logarithmic function
The function f(x) = log3x has domain x > 0. Find f−1(x) and state its domain.
Step 1: Set y = log₃ x and swapy = log₃ x → swap: x = log₃ yStep 2: Rearrange to solve for ylog₃ y = x ⟺ y = 3ˣStep 3: Domain of f⁻¹ = range of f = ℝf⁻¹(x) = 3ˣ, domain x ∈ ℝlog and exponential of the same base are perfect inverses — always
WE 5
Evaluate a log using the change of base formula
Without a calculator, evaluate log8 32 using the change of base formula.
Convert to base 2 (since both 8 and 32 are powers of 2)log₈ 32 = log₂ 32 / log₂ 8= 5 / 3log₈ 32 = 5/3picking a useful base (where the numbers simplify cleanly) makes this much faster than going through ln
WE 6
Identify a function from its key features
A function passes through (0, 1) and (2, 9), and has a horizontal asymptote at y = 0. Determine the equation of the function in the form y = ax.
Step 1: Identify the type — y-intercept (0, 1) and HA y = 0 → exponentialStep 2: Use (2, 9) to find a9 = a²a = 3 (must be positive)y = 3ˣthe (1, a) anchor point also confirms — at x = 1 the function gives y = 3 ✓
💡 Top tips
(0, 1) for exponentials, (1, 0) for logs. Memorise these two — they instantly fix the orientation of the graph.
Inverses of each other: log undoes exponential and vice versa. Use this to solve equations like 2x = 11 → x = log2 11.
Reflection in y = x means features swap: y-intercept ↔ x-intercept, horizontal asymptote ↔ vertical asymptote.
Convert to e for calculus: ax = ex ln a. Essential when you start differentiating later.
Change of base: logax = ln x / ln a. Use it to evaluate logs of any base on your GDC.
log of 1 is 0, regardless of base. log of the base itself is 1 (e.g. log5 5 = 1).
Negative or zero inputs to log → undefined. Always check the domain of any log expression you build.
⚠ Common mistakes
Forgetting that log requires x > 0. log of zero or a negative is undefined — affects domain.
Mixing up the asymptotes: exponentials have horizontal (y = 0), logs have vertical (x = 0).
Confusing logax with 1/ax. Logs are inverses of exponentials, not reciprocals.
Using log when you should use ln (or vice versa). They share all log laws — just keep the bases consistent.
Missing the y-intercept of an exponential. Every y = ax passes through (0, 1) — always.
Forgetting a > 0 in y = ax. Negative bases cause undefined values for non-integer exponents.
Substituting wrongly into change of base: logax = ln x / ln a — the input goes on top, the base on the bottom.
Now you’ve got the building blocks. The next note covers solving equations analytically — using the inverse relationship to turn exponential and logarithmic equations into something solvable. After that, the focus shifts to graphical solving when algebra runs out of moves.
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