Logarithmic functions and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

The graph of $y = 2^x$ passes through $(0, 1)$ , $(1, 2)$ and $(3, 8)$ . Swap the coordinates of each point.

Which function has a graph through the swapped points?

Key idea

$y = \log_b x$ is the inverse of $y = b^x$ : its graph is the reflection of $y = b^x$ in the line $y = x$ .

The graph of

y = \log_b x

passes through

(1, 0)

and

(b, 1)

, has domain

x > 0

and a vertical asymptote

x = 0

For $y = \log_b(x - h)$ , everything shifts $h$ units right: the asymptote is $x = h$ and the graph crosses the $x$ -axis where $x - h = 1$ .

Worked example

Where does $y = \log_2(x - 3)$ cross the $x$ -axis?

\log_2(x - 3) = 0

On the x-axis, y = 0.

x - 3 = 1

A logarithm is 0 when its input is 1.

x = 4

Add 3.

Your turn, guided

Your turn: where does $y = \ln(x - 5)$ cross the $x$ -axis? Fill in the missing line.

x - 5 = 1

ln of the input is 0 when the input is 1.

x = 6

Add 5.

Practice ★

Which statements are true about the graph of $y = \log_{ 9 } x$ ? Tick all that are true.
1. It passes through (1, 0).
2. It passes through (9, 1).
3. It passes through (0, 1).
4. It has the horizontal asymptote y = 0.
5. It is defined only for x > 0.
Which statements are true about the graph of $y = \log_{ 2 } x$ ? Tick all that are true.
1. It passes through (1, 0).
2. It passes through (2, 1).
3. It passes through (0, 1).
4. It has the horizontal asymptote y = 0.
5. It is defined only for x > 0.

The graph of $y = \log_b x$ passes through the point $(9, 2)$ . What is $b$ ?
The graph of $y = \log_b x$ passes through the point $(81, 4)$ . What is $b$ ?
The graph of $y = \log_b x$ passes through the point $(8, 3)$ . What is $b$ ?
The graph of $y = \log_b x$ passes through the point $(625, 4)$ . What is $b$ ?
The graph of $y = \log_b x$ passes through the point $(64, 3)$ . What is $b$ ?

Watch out!

Practice ★★

Here is the graph of $y = \log_{ 3 }(x - 5)$ . Tap the point where it crosses the $x$ -axis.
Here is the graph of $y = \log_{ 2 }(x - 4)$ . Tap the point where it crosses the $x$ -axis.

What is the domain of $y = \ln(x - 6)$ ? Give it as an interval.
What is the domain of $y = \ln(x - 9)$ ? Give it as an interval.
What is the domain of $y = \ln(x - 7)$ ? Give it as an interval.
What is the domain of $y = \ln(x - 2)$ ? Give it as an interval.
What is the domain of $y = \ln(x - 8)$ ? Give it as an interval.

Exit ticket

Three questions to check you have got it.

Where does the graph of $y = \ln(x - 3)$ cross the $x$ -axis? Give the value of $x$ .
Where does the graph of $y = \ln(x - 6)$ cross the $x$ -axis? Give the value of $x$ .
Where does the graph of $y = \ln(x - 7)$ cross the $x$ -axis? Give the value of $x$ .

Summary card

Logarithmic functions

$y = \log_b x$ is the inverse of $y = b^x$ .
Through $(1, 0)$ and $(b, 1)$ ; domain $x > 0$ ; asymptote $x = 0$ .
$y = \log_b(x - h)$ : asymptote $x = h$ ; crosses the $x$ -axis at $x = h + 1$ .

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