SAMath MethodsSyllabus dot point

What does the graph of a logarithmic function look like and how is it related to the exponential graph?

The logarithmic graph is the reflection of the exponential graph in the line y = x, with a vertical asymptote and a characteristic slow growth.

The shape, domain, asymptote and intercepts of y = log_a(x), how it reflects the exponential graph, and how transformations shift and stretch it.

Generated by Claude Opus 4.77 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
Key features of $y=\log_a x$ ($a>1$)
Reflection of the exponential
Transformations
Common errors
Why it matters

What this dot point is asking

The logarithmic function is the inverse of the exponential function, so its graph is the exponential graph reflected in the line $y=x$ . Knowing the basic shape lets you sketch any transformed log curve.

Key features of $y=\log_a x$ ( $a>1$ )

The point $(a,1)$ also lies on the curve, since $\log_a a=1$ - a handy second point when sketching.

Reflection of the exponential

The exponential $y=a^x$ has a horizontal asymptote ( $y=0$ ), domain all reals, and range $y>0$ . Reflecting in $y=x$ swaps these:

Feature	$y=a^x$	$y=\log_a x$
Domain	all $x$	$x>0$
Range	$y>0$	all $y$
Asymptote	$y=0$ (horizontal)	$x=0$ (vertical)
Key point	$(0,1)$	$(1,0)$

This is exactly the inverse relationship: every point $(p,q)$ on $y=a^x$ becomes $(q,p)$ on $y=\log_a x$ .

Transformations

A transformed log function combines shifts and stretches. For $y=\log_a(x-h)+k$ :

$h$ shifts the graph (and its asymptote) horizontally to $x=h$ .
$k$ shifts the graph vertically by $k$ .

Sketching a transformed log curve

Worked example

Describe the key features of $y=\ln(x-2)+1$ .

Start from $y=\ln x$ (asymptote $x=0$ , passes through $(1,0)$ ).

The $(x-2)$ shifts everything right by 2, so the asymptote is $x=2$ and the domain is $x>2$ .
The $+1$ shifts everything up by 1.

The point $(1,0)$ on $y=\ln x$ maps to $(1+2,\,0+1)=(3,1)$ .

x-intercept (where $y=0$ ): $\ln(x-2)+1=0\Rightarrow \ln(x-2)=-1\Rightarrow x-2=e^{-1}\Rightarrow x=2+e^{-1}\approx 2.37$ .

So the curve rises from the asymptote at $x=2$ , crosses the x-axis near $x\approx 2.37$ , and passes through $(3,1)$ .

Common errors

Why it matters

Recognising the log graph's shape, domain and asymptote is examined directly and underpins your interpretation of logarithmic models (such as decibel, pH and Richter scales). It also reinforces the inverse relationship with exponentials that drives the equation-solving in this topic.

Exam-style practice questions

Practice questions written in the style of SACE Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2017 SACE Stage 22 marksConsider the graphs of f(x) = ln(5x) and g(x) = ln(x). Point A is at (2, f(2)) and point B is at (2, g(2)). What is the exact distance between points A and B?

Show worked answer →

A and B share the same x-coordinate (x = 2), so the distance between them is simply the vertical gap |f(2) - g(2)|.

f(2) = ln(5 times 2) = ln 10, and g(2) = ln 2.

Distance = ln 10 - ln 2 = ln(10 / 2) = ln 5 (using the quotient law).

So the exact distance is ln 5 units. Marks: one for evaluating both heights, one for combining with the quotient law to the exact value ln 5. Because A is directly above B, no horizontal component enters the distance. Leaving the answer as ln 5 (exact) rather than 1.61 is required.

2017 SACE Stage 21 marksComment on the relationship between the graphs of g(x) = ln(x) and h(x) = ln(kx), where k > 0.

Show worked answer →

Use the product law to split h(x): h(x) = ln(kx) = ln k + ln x = g(x) + ln k.

So h(x) is obtained from g(x) by adding the constant ln k. Graphically, the graph of h is the graph of g translated vertically by ln k units (upwards if k > 1, downwards if 0 < k < 1).

The single mark is for identifying the relationship as a vertical translation of g(x) by ln k. The key insight is that a horizontal scale factor inside a logarithm becomes an additive constant, which is a vertical shift, not a horizontal stretch.