What are exponential functions and how are they used in IB DP Mathematics?

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent, typically written as f(x) = a^x, where 'a' is a positive constant. In IB DP Mathematics, these functions are used to model real-world phenomena such as population growth, radioactive decay, and compound interest, providing students with tools to analyze and interpret exponential growth and decay in various contexts.

How do logarithmic functions relate to exponential functions in the IB DP curriculum?

Logarithmic functions are the inverse of exponential functions. If y = a^x is an exponential function, then x = log_a(y) is the corresponding logarithmic function. In the IB DP curriculum, understanding this relationship helps students solve equations involving exponential growth and decay, and it is essential for topics such as solving exponential equations and understanding the properties of logarithms.

What are the key properties of logarithms that IB DP students should know?

IB DP students should be familiar with several key properties of logarithms: the product rule (log_a(xy) = log_a(x) + log_a(y)), the quotient rule (log_a(x/y) = log_a(x) - log_a(y)), and the power rule (log_a(x^b) = b*log_a(x)). These properties are crucial for simplifying logarithmic expressions and solving logarithmic equations, which are common tasks in the IB DP Mathematics curriculum.

How can students solve exponential equations using logarithms in IB DP Mathematics?

To solve exponential equations using logarithms, students can take the logarithm of both sides of the equation. For example, if the equation is a^x = b, taking the logarithm base 'a' of both sides gives x = log_a(b). This method allows students to isolate the variable and find its value, which is a key skill in the IB DP Mathematics curriculum for solving real-world problems involving exponential growth and decay.

What is the significance of the natural logarithm in IB DP Mathematics?

The natural logarithm, denoted as ln(x), is a logarithm with base 'e', where 'e' is an irrational constant approximately equal to 2.718. In IB DP Mathematics, the natural logarithm is significant because it frequently appears in calculus, particularly in the differentiation and integration of exponential functions. Understanding the natural logarithm is essential for students to solve complex mathematical problems and to model continuous growth processes.

Why is "Writing $\log(x + y) = \log x + \log y$." a common mistake in Exponential and Logarithmic Functions?

Why it happens: Over-applying product rule. How to avoid it: Product rule applies to (xy) only. (x + y) has no simplification.

Why is "Using $T = \ln 2 / k$ without the absolute value." a common mistake in Exponential and Logarithmic Functions?

Why it happens: Carelessness about sign of k. How to avoid it: Half-life requires k < 0 (decay). The formula uses |k| so the time is positive.

Why is "Using $N = N_0 e^{rt}$ when the question says 'annually compounded'." a common mistake in Exponential and Logarithmic Functions?

Why it happens: Defaulting to continuous. How to avoid it: 'Annually' / 'each year' ⇒ discrete P(1 + r)^t. 'Continuously' ⇒ P e^rt.

Exponential and Logarithmic FunctionsIB Maths AA HL

Exponential and Logarithmic Functions

Q: Why is "Using $N = N_0 e^{rt}$ when the question says 'annually compounded'." a common mistake in Exponential and Logarithmic Functions?

Why it happens: Defaulting to continuous. How to avoid it: 'Annually' / 'each year' ⇒ discrete P(1 + r)^t. 'Continuously' ⇒ P e^rt.

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Exponential and Logarithmic Functions — IB Maths AA HL: $e^x$, $\ln x$, full log laws and HL applications

Mastery of exponentials and logarithms is the gateway to HL calculus. This note covers laws of logs, change of base, exponential and logarithmic equations, growth/decay modelling, and graphical features for HL Paper 1 and Paper 3.

What you’ll learn

Mapped to the IB DP Maths AA HL subject guide (2021 onwards (applies to 2026 exams)).

AO1 — State and apply log laws and change-of-base.
AO1 — Recognise the natural exponential and logarithm.
AO2 — Solve exponential and logarithmic equations using logs and substitution.
AO2 — Apply continuous growth/decay models, including half-life.
AO3 — State domain restrictions explicitly.
AO4 — Use GDC for Paper 2 numerical answers.

Log laws and change of base

Booklet formulas — fluency required.

For $a > 0$ , $a \ne 1$ , and positive $x, y$ :

Product: $\log_a(xy) = \log_a x + \log_a y$
Quotient: $\log_a(x/y) = \log_a x - \log_a y$
Power: $\log_a(x^k) = k \log_a x$
Change of base: $\log_a x = \dfrac{\log_b x}{\log_b a}$

Special values:

$\log_a 1 = 0$
$\log_a a = 1$
$\ln e = 1$
$a^{\log_a x} = x$
$\log_a(a^x) = x$

Worked example. Express $\ln 18 + \ln 4 - \ln 8$ as a single logarithm.

$= \ln(18 \cdot 4 / 8) = \ln 9 = 2 \ln 3$ .

Worked example (Paper 1 exact form). Solve $\log_2 x + \log_2 (x + 2) = 3$ .

Combine: $\log_2[x(x + 2)] = 3 \Rightarrow x(x + 2) = 8 \Rightarrow x^2 + 2x - 8 = 0 \Rightarrow (x - 2)(x + 4) = 0$ .

Domain ( $x > 0$ ): reject $x = -4$ , accept $x = 2$ .

Combine logs only when both arguments are positive.
Change of base for non-standard bases.
Always state and apply the domain restriction.

Exponential equations — five techniques

Same base, take logs, substitute, factor, GDC.

Technique 1: same base. If both sides reduce to the same base, equate exponents.

Worked example. Solve $9^x = 27^{x - 1}$ .

$(3^2)^x = (3^3)^{x-1} \Rightarrow 2x = 3(x-1) \Rightarrow x = 3$ .

Technique 2: take logs. Different bases — take $\ln$ .

Worked example. Solve $3 \cdot 5^x = 100$ .

$5^x = 100/3 \Rightarrow x \ln 5 = \ln(100/3) \Rightarrow x = \dfrac{\ln(100/3)}{\ln 5} \approx 2.180$ .

Technique 3: substitution. Hidden quadratics, as covered in Equations subtopic.

Worked example. Solve $4^x - 5 \cdot 2^x + 4 = 0$ .

Note $4^x = (2^x)^2$ . Let $u = 2^x$ : $u^2 - 5u + 4 = 0 \Rightarrow (u - 1)(u - 4) = 0$ .

$u = 1$ : $x = 0$ . $u = 4$ : $x = 2$ .

Technique 4: factor. Sometimes terms share a factor.

Worked example. Solve $2^{x+2} - 2^x = 24$ .

$2^x(2^2 - 1) = 24 \Rightarrow 3 \cdot 2^x = 24 \Rightarrow 2^x = 8 \Rightarrow x = 3$ .

Technique 5: GDC graphical (Paper 2 only).

$y = e^x$ and $y = \ln x$ are inverse functions, so their graphs are mirror images in $y = x$. Each has the other's asymptote as its limiting axis.

Match bases when possible.
Take $\ln$ when bases differ.
Substitute for hidden quadratics.
Factor when terms share a common power.

Continuous growth and decay

$N(t) = N_0 e^{kt}$ .

Continuous model: $N(t) = N_0 e^{kt}.$

$N_0$ = initial value, $k$ = continuous rate. $k > 0$ : growth. $k < 0$ : decay.

Half-life (for decay): $T = \dfrac{\ln 2}{|k|}$ .

Doubling time (for growth): $T = \dfrac{\ln 2}{k}$ .

Linearisation. Taking $\ln$ of $N = N_0 e^{kt}$ gives $\ln N = \ln N_0 + kt$ — LINEAR in $t$ . This is the basis of fitting exponential models to data (linear regression on $\ln N$ vs $t$ ).

Worked example (decay). A radioactive isotope has half-life 12 years. Find $k$ , then how long until 90% has decayed.

$T = 12 \Rightarrow k = -\dfrac{\ln 2}{12} \approx -0.0578$ .

90% decayed ⇒ 10% remains: $N_0 e^{kt} = 0.1 N_0 \Rightarrow kt = \ln 0.1 = -\ln 10$ .

$t = \dfrac{-\ln 10}{-\ln 2 / 12} = \dfrac{12 \ln 10}{\ln 2} \approx 39.86$ years.

Worked example (compound interest comparison). $5000 at $4.5\%$ . Compare end-of-10-year value if compounded (a) annually, (b) continuously.

(a) $A = 5000(1.045)^{10} \approx \$ 7765$.

(b) $A = 5000 e^{0.045 \cdot 10} \approx 5000 e^{0.45} \approx \$ 7841$.

Continuous compounding gives slightly more — the limiting case as compounding frequency goes to infinity.

Newton's law of cooling. $T(t) - T_{\text{room}} = (T_0 - T_{\text{room}}) e^{-kt}$ — applied physics problem on Paper 3.

$N(t) = N_0 e^{kt}$ : growth ( $k > 0$ ) or decay ( $k < 0$ ).
Half-life $= \ln 2 / |k|$ ; doubling time $= \ln 2 / k$ .
$\ln N$ vs $t$ is LINEAR — basis of data fitting.
Continuous compound interest $A = P e^{rt}$ .

How it’s examined

Paper 1: exact solving, log law manipulation, domain rejection. Paper 2: GDC modelling. Paper 3: extended growth/decay with parameter estimation. Examiner reports stress the domain restriction step when combining logs.

Sources: IB Diploma Programme Mathematics: Analysis and Approaches subject guide (IBO, official). Last reviewed 2026-05-31.

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Step-by-step worked examples — Exponential and Logarithmic Functions

Step-by-step solutions to past-paper-style questions on exponential and logarithmic functions, written exactly the way a tutor would explain them at the board.

1Simplify with log laws
Getting started• log laws
Question
Express $\ln 50 - \ln 5 + 2\ln 3$ as a single logarithm.
Step-by-step solution
1. Step 1
  $\ln 50 - \ln 5 = \ln 10$ .
2. Step 2
  $2\ln 3 = \ln 9$ .
3. Step 3
  Sum: $\ln 10 + \ln 9 = \ln 90$ .
Answer
$\ln 90$ .
2Same-base exponential
Building confidence• exponential
Question
Solve $8^{x+1} = 4^{2x - 1}$ .
Step-by-step solution
1. Step 1
  $8 = 2^3$ and $4 = 2^2$ : $2^{3(x+1)} = 2^{2(2x-1)}$ .
2. Step 2
  Equate exponents: $3x + 3 = 4x - 2 \Rightarrow x = 5$ .
Answer
$x = 5$ .
3Hidden quadratic in $e^x$
Stretch• substitution
Question
Solve $e^{2x} - 6e^x + 5 = 0$ .
Step-by-step solution
1. Step 1
  Let $u = e^x$ : $u^2 - 6u + 5 = 0 \Rightarrow (u - 1)(u - 5) = 0$ .
2. Step 2
  Both positive, both valid: $u = 1 \Rightarrow x = 0$ ; $u = 5 \Rightarrow x = \ln 5$ .
Answer
$x = 0$ or $x = \ln 5$ .
4Continuous decay — find time
Stretch• decay, application
Question
A sample decays continuously with half-life 8 days. Find $k$ , then the time for 75% to decay.
Step-by-step solution
1. Step 1
  Half-life: $T = \ln 2 / |k| = 8 \Rightarrow k = -\ln 2 / 8 \approx -0.0866$ .
2. Step 2
  75% decayed ⇒ 25% remains: $N_0 e^{kt} = 0.25 N_0 \Rightarrow kt = \ln 0.25 = -\ln 4$ .
3. Step 3
  $t = \ln 4 / (\ln 2 / 8) = 8 \ln 4 / \ln 2 = 16$ days.
Answer
$16$ days (exactly two half-lives).

Key Definitions and Keywords — Exponential and Logarithmic Functions

Definitions to memorise and the exact keywords mark schemes credit for exponential and logarithmic functions answers — sharpened from recent examiner reports for the 2026 IB DP Maths AA HL sitting.

Logarithm $\log_a y$
Examiner keyword
The exponent $x$ such that $a^x = y$ . Defined for $a > 0$ , $a \ne 1$ , $y > 0$ .
Natural logarithm $\ln$
Examiner keyword
$\ln = \log_e$ , with $e \approx 2.71828$ .
Half-life
Time for decaying quantity to halve. $T = \ln 2 / |k|$ for $N_0 e^{kt}$ with $k < 0$ .

Common Mistakes and Misconceptions — Exponential and Logarithmic Functions

The traps other students keep falling into on exponential and logarithmic functions questions — taken from recent IB DP Maths AA HL examiner reports and mark schemes — and how to avoid them.

✕Writing $\log(x + y) = \log x + \log y$ .
Why it happens
Over-applying product rule.
How to avoid it
Product rule applies to $\log(xy)$ only. $\log(x + y)$ has no simplification.
✕Using $T = \ln 2 / k$ without the absolute value.
Why it happens
Carelessness about sign of $k$ .
How to avoid it
Half-life requires $k < 0$ (decay). The formula uses $|k|$ so the time is positive.
✕Using $N = N_0 e^{rt}$ when the question says 'annually compounded'.
Why it happens
Defaulting to continuous.
How to avoid it
'Annually' / 'each year' ⇒ discrete $P(1 + r)^t$ . 'Continuously' ⇒ $P e^{rt}$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Exponential and Logarithmic Functions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Exponential and Logarithmic Functions

Short Study Notes in the form of Slides

Short Study Notes — Exponential and Logarithmic Functions

Detailed Notes

What you’ll learn

How it’s examined

1Simplify with log laws

2Same-base exponential

3Hidden quadratic in exe^xex

4Continuous decay — find time

Logarithm log⁡ay\log_a yloga​y

Natural logarithm ln⁡\lnln

Half-life

✕Writing log⁡(x+y)=log⁡x+log⁡y\log(x + y) = \log x + \log ylog(x+y)=logx+logy.

✕Using T=ln⁡2/kT = \ln 2 / kT=ln2/k without the absolute value.

✕Using N=N0ertN = N_0 e^{rt}N=N0​ert when the question says 'annually compounded'.

Practice questions

Past paper style quiz

Exponential and Logarithmic Functions — frequently asked questions

3Hidden quadratic in $e^x$

Logarithm $\log_a y$

Natural logarithm $\ln$

✕Writing $\log(x + y) = \log x + \log y$ .

✕Using $T = \ln 2 / k$ without the absolute value.

✕Using $N = N_0 e^{rt}$ when the question says 'annually compounded'.