Exponentials & Logarithms (AA HL)
This topic looks at exponential and logarithmic functions as graphs rather than just algebraic expressions: what shape they take, where their asymptotes sit, how their domains are restricted, and how the two families reflect each other as inverse functions. It builds directly on the laws of exponents and logarithms, applying them to sketching, transforming and solving.
What the syllabus says
This topic maps onto two points in the official IB Analysis & Approaches syllabus, both core SL content that's also examined at HL.
| Code | Syllabus content |
|---|---|
| SL2.9 | Exponential functions and their graphs: \(f(x)=a^x,\ a>0\), \(f(x)=e^x\). Logarithmic functions and their graphs: \(f(x)=\log_a x,\ x>0\), \(f(x)=\ln x,\ x>0\). Exponential and logarithmic functions as inverses of each other, with \(a^x=e^{x\ln a}\). |
| SL2.10 | Solving equations, both graphically and analytically, including exponential equations. Use of technology to solve equations with no convenient analytic approach. |
These build directly on the laws of exponents and logarithms (SL1.5, SL1.7) and on finding an inverse function (AHL2.14).
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is an exponential function?
An exponential function has the form \(f(x)=a^x\) for a fixed base \(a>0\), or \(f(x)=e^x\) using Euler's number. Unlike a polynomial, the variable is in the exponent, which makes the function grow (or decay) increasingly fast.
e.g. \(f(x)=2^x\): \(f(3)=2^3=8\).
What is a logarithmic function?
A logarithmic function, \(f(x)=\log_a x\) or \(f(x)=\ln x\), is the inverse of an exponential function. Its graph is a reflection of the corresponding exponential graph in the line \(y=x\).
e.g. \(f(x)=\ln x\): \(f(1)=0\), since \(e^0=1\).
What is the horizontal asymptote of an exponential graph?
An exponential graph \(f(x)=a^x+c\) always approaches (but never reaches) the horizontal line \(y=c\) on one side, since \(a^x\to0\) as \(x\to-\infty\) (for \(a>1\)).
e.g. \(f(x)=3e^x-1\): as \(x\to-\infty\), \(f(x)\to-1\), so the horizontal asymptote is \(y=-1\).
What is the domain of a logarithmic function?
A logarithmic function is only defined when its argument is strictly positive, so its domain is always restricted. For \(f(x)=\log_a(x-h)\), you need \(x-h>0\).
e.g. \(f(x)=\ln(x-2)\) needs \(x-2>0\), so the domain is \(x>2\).
What is the inverse of an exponential function?
The inverse of an exponential function is found by isolating the exponential term and taking a logarithm of both sides. Because exponentials and logarithms are inverses of each other, the result is always logarithmic.
e.g. \(f(x)=3e^x-1\) has inverse \(f^{-1}(x)=\ln\!\left(\dfrac{x+1}{3}\right)\), since \(e^x=\dfrac{y+1}{3}\Rightarrow x=\ln\dfrac{y+1}{3}\).
Key formulas
Two facts connect every exponential and logarithmic function on this topic. The two tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The exponential-logarithm equivalence and the \(a^x=e^{x\ln a}\) identity are both given in the formula booklet; asymptote and domain shifts are read directly from the transformed function rather than looked up.
| Formula | Used for | Booklet? |
|---|---|---|
| \(a^x=b\iff\log_a b=x\) | Exponential-logarithm equivalence | ✓ Yes |
| \(a^x=e^{x\ln a}\) | Rewriting any exponential in base \(e\) | ✓ Yes |
| \(f(x)=a^x+c\Rightarrow\) asymptote \(y=c\) | Horizontal asymptote after a vertical shift | Not in booklet |
| \(f(x)=\log_a(x-h)\Rightarrow\) domain \(x>h\) | Domain after a horizontal shift | Not in booklet |
Exponential vs logarithmic functions
As inverses of each other, these two function families mirror one another in almost every feature.
| Feature | Exponential \(f(x)=a^x\) | Logarithmic \(f(x)=\log_a x\) |
|---|---|---|
| Domain | All real \(x\) | \(x>0\) |
| Range | \(y>0\) | All real \(y\) |
| Asymptote | Horizontal, \(y=0\) | Vertical, \(x=0\) |
| Passes through | \((0,1)\) | \((1,0)\) |
| Relationship | Reflections of each other in \(y=x\) | |
Exponential functions
Every exponential graph shares the same core shape - only its steepness and starting point change.
General form
\[f(x)=a\cdot b^{x}+c\]
\(b\) controls growth (\(b>1\)) or decay (\(0
Always positive (before shifting)
\(a^x>0\) for every real \(x\) - the graph never touches or crosses the \(x\)-axis unless it's been shifted.
Horizontal asymptote
Approaches \(y=c\) as \(x\to-\infty\) (growth) or \(x\to+\infty\) (decay), for \(f(x)=a\cdot b^x+c\).
Solving exponential and logarithmic equations
Most equations on this topic reduce to one of these three moves.
Isolate first
Get the exponential or logarithmic term completely alone on one side before taking logs or exponentiating.
Take logs of both sides
Once isolated, apply \(\ln\) (or any convenient base) to both sides to bring the variable out of the exponent.
Use technology when needed
Some equations, like \(e^x=\sin x\), have no algebraic solution - graph both sides and find the intersection instead.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Let \(f(x)=3e^{x}-1\).
(a) Find \(f(0)\).
(b) State the equation of the horizontal asymptote of \(y=f(x)\).
(c) Find \(f^{-1}(x)\).
Worked solution
(a) \(f(0)=3-1=2.\) A1
(b) As \(x\to-\infty,\ f\to-1,\) so \(y=-1.\) A1
(c) \(y=3e^x-1\Rightarrow e^x=\dfrac{y+1}{3}\Rightarrow f^{-1}(x)\) M1
\(=\ln\!\left(\dfrac{x+1}{3}\right).\) A1
\(f(x)=\ln(x-2)+1.\)
(a) State the domain.
(b) State the vertical asymptote.
(c) Find the \(x\)-intercept exactly.
Worked solution
(a) \(\ln(x-2)\) needs \(x-2>0\), so the domain is \(x>2.\) A1
(b) The log blows up as its argument \(\to0^+\), giving a vertical asymptote \(x=2.\) A1
(c) Set \(f(x)=0:\ \ln(x-2)=-1\Rightarrow x-2=e^{-1}\Rightarrow x\) M1
\(=2+e^{-1}.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting the horizontal asymptote shifts with the function. \(f(x)=e^x+2\) has asymptote \(y=2\), not \(y=0\) - a common slip is quoting the asymptote of the un-shifted function instead of the given one.
- Taking logs before fully isolating the exponential term. You must get the exponential completely alone on one side - \(\log\) doesn't distribute over addition, so \(\ln(2e^x+3)\) can't be split into separate pieces.
- Ignoring the domain restriction of a logarithm. \(\ln(x-2)\) needs \(x>2\), not \(x>0\) - the restriction applies to whatever is inside the log, after any shift.
- Mixing up which asymptote belongs to which function after reflecting in \(y=x\). An exponential's horizontal asymptote becomes a vertical asymptote for its logarithmic inverse, at the same numerical value.
Using your GDC
There isn't a single fixed button sequence for this topic, since it spans both no-calculator algebra and calculator-assisted graphing.
On Paper 1, finding a domain, an asymptote, or an inverse function is done analytically - a GDC won't help with that reasoning. On Paper 2, though, your calculator becomes genuinely useful for solving exponential and logarithmic equations that are awkward or impossible to rearrange by hand: graph both sides of the equation and use the intersection tool, or enter the equation directly into an equation solver. Evaluating a logarithm in any base, or a value like \(e^{2.336}\), is also just a direct calculator computation once you've set the expression up algebraically.
See the full GDC guide for calculator-specific graphing and equation-solving steps.
Ready to practise properly?
Exponentials & logarithms questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
Why does an exponential graph have a horizontal asymptote?
Because \(a^x\) gets closer and closer to zero as \(x\) becomes very negative (for \(a>1\)), but never actually reaches it. Adding a constant, as in \(f(x)=a^x+c\), shifts that asymptote to \(y=c\).
Why do logarithmic functions have a restricted domain?
Because a logarithm is only defined for a positive argument. For \(f(x)=\log_a(x-h)\), you need \(x-h>0\), so the domain is \(x>h\) - trying to take the log of zero or a negative number has no real value.
How do I find the inverse of an exponential function?
Swap \(x\) and \(y\), then isolate the exponential term and take logarithms of both sides to solve for \(y\). Since exponential and logarithmic functions are inverses of each other, the inverse of an exponential is always a logarithmic function.
Can I use my GDC to solve exponential and logarithmic equations?
Yes, on Paper 2 - graphing both sides of an equation and finding the intersection, or using an equation solver, handles equations that would be awkward or impossible to solve by hand. Paper 1 questions on domain, asymptotes, and finding an inverse are done analytically, without a calculator. See the GDC guide for model-specific instructions.
Sub-topics
Exponentials & Logarithms broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA HL syllabus unit, in case you want to keep going.