Exponential & Log Graphs (AA HL)
The graphs of \(f(x)=a^x\) and \(f(x)=\log_a x\) are two of the most exam-tested shapes in the AA syllabus, and once you can picture how they behave as inverses of each other, questions about domain, range and asymptotes become routine. This page focuses on reading and sketching those shapes, plus the transformations examiners like to layer on top. It's part of the broader Exponentials & Logarithms topic.
11 questions on this sub-topic.
How the two graphs connect
Covered under IB syllabus reference SL2.9: exponential functions \(f(x)=a^x\), \(a>0\), and \(f(x)=e^x\); logarithmic functions \(f(x)=\log_a x\), \(x>0\), and \(f(x)=\ln x\), \(x>0\); and exponential and logarithmic functions as inverses of each other.
Exponential-logarithm equivalence
\(a^x=b\iff\log_a b=x\). This is exactly what makes the two graphs inverses - reflecting one in the line \(y=x\) produces the other.
Rewriting in base \(e\)
Any exponential can be written \(a^x=e^{x\ln a}\), which is how your GDC evaluates and graphs exponentials with a base other than \(e\).
Need the full syllabus wording, the formula reference table, or GDC-specific steps? See Exponentials & Logarithms.
Worked examples
\(f(x)=e^{x}.\)
(a) State the range.
(b) Sketch \(y=e^x\), marking the intercept and asymptote.
Worked solution
(a) \(e^x>0\) for every real \(x\), so the range is \(f(x)>0.\) A1
(b) The curve passes through \((0,1)\) A1
is increasing throughout A1
with horizontal asymptote \(y=0\) as \(x\to-\infty.\) A1 (See sketch.)
\(f(x)=e^x.\) The graph of \(g(x)=-2e^{x-1}\) is obtained from the graph of \(f\) by a sequence of transformations.
(a) Describe the transformations, in order.
(b) State the range of \(g.\)
(c) State the equation of the horizontal asymptote of \(g.\)
Worked solution
(a) Replacing \(x\) by \(x-1\) is a translation \(1\) unit right; B1 multiplying by \(2\) is a vertical stretch, scale factor \(2\); B1 and the negative sign is a reflection in the \(x\)-axis. B1
(b) Since \(e^{x-1}>0\) for all \(x,\) \(-2e^{x-1}<0\) for all \(x.\) R1 Range: \(g(x)<0.\) A1
(c) Vertical translations and reflections do not move the horizontal asymptote from \(y=0.\) A1
Common mistakes
- 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 - not the same axis.
- 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, so a translated log graph doesn't just start at the origin.
- Applying a vertical translation to the asymptote when it isn't a vertical translation. Only adding or subtracting a constant outside the exponential shifts \(y=0\) up or down; a stretch or reflection alone leaves it exactly where it was.
Ready to practise properly?
11 exponential and log graph questions, marked instantly like the real exam.
Quick answers
What is the range and asymptote of \(y = e^x\)?
\(y = e^x\) has range \(f(x) > 0\) and a horizontal asymptote \(y = 0\) as \(x\to-\infty\); the curve passes through \((0,1)\) and increases throughout.
How are exponential and logarithmic graphs related?
They are inverse functions of each other, so their graphs are reflections of one another in the line \(y = x\) - which is also why the horizontal asymptote of an exponential graph becomes the vertical asymptote of its logarithmic inverse, at the same numerical value.