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Möbius Mathematics Readiness

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domain $ \{x \mid x \gt 0,~x \in \mathbb{R} \} $ and range $\{y \mid y \in \mathbb{R} \} $,
an $x $-intercept at $(1, 0) $ and no $ y $-intercept, and
a vertical asymptote of $x = 0 $ (the $y$-axis).

Observations on the Value of $c$

Check Your Understanding D

Given the graphs of $3$ logarithmic functions of the form $y = {log}_{c} (x)$ , match each equation to its corresponding graph by typing the appropriate number into the space beside each equation.

$y = {log}_{\frac{2}{3}} (x)$
$y = {log}_{\frac{1}{4}} (x)$
$y = {log}_{6} (x)$

Feedback:

For logarithmic functions of the form $y = {log}_{c} (x)$ :

When $c > 1$ the graph is increasing. The smaller the value of $c$ (closer to $1$ ), the faster the graph increases (steeper positive slope).
When $0 < c < 1$ the graph is decreasing. The larger the value of $c$ (closer to $1$ ), the faster the graph decreases (steeper negative slope).

Thus, $y = {log}_{\frac{1}{4}} (x)$ and $y = {log}_{\frac{2}{3}} (x)$ are decreasing functions, with $y = {log}_{\frac{2}{3}} (x)$ steeper than $y = {log}_{\frac{1}{4}} (x)$ , and $y = {log}_{6} (x)$ is an increasing function.

If $ \textcolor{BrickRed}{a}< 0, ~y = \log_{c}(x)$ is reflected in the $ x $-axis.
$ y = \log_{c}(x) $ is stretched vertically about the $ x $-axis by a factor of $\lvert \textcolor{BrickRed}{a} \rvert$.
If $\textcolor{NavyBlue}{b} < 0, ~y = \log_{c}(x) $ is reflected in the $ y$-axis.
$y = \log_{c}(x) $ is stretched horizontally about the $ y $-axis by a factor of $ \dfrac{1}{\lvert \textcolor{NavyBlue}{b} \rvert} $.
$y = \log_{c}(x) $ is translated horizontally $ \textcolor{Mulberry}{h} $ units.

$y = \log_{c}(x) $ is translated vertically $ \textcolor{Violet}{k} $ units.

Check Your Understanding E

What transformations are applied to $y = {log}_{3} (x)$ to obtain $y = \frac{1}{3} \log_{3} (- \frac{1}{2} x - \frac{3}{2}) - 2$ ?

a) A reflection in the:

$x$ -axis

$y$ -axis

both axes

no reflection

b) A vertical stretch about the $x$ -axis by a factor of

c) A horizontal stretch about the $y$ -axis by a factor of

d) A horizontal translation unit(s)

		left
		right

e) A vertical translation unit(s)

		up
		down

Feedback:

The general transformed logarithmic function is $f (x) = a {log}_{3} (b (x - h)) + k$ .

Expressing the function $y = \frac{1}{3} \log_{3} (- \frac{1}{2} x - \frac{3}{2}) - 2$ in this form, we have $y = \frac{1}{3} \log_{3} (- \frac{1}{2} (x + 3)) - 2$ , where $a = \frac{1}{3}$ , $b = - \frac{1}{2}$ , $h = -3$ , $k = -2$ .

If $a < 0$ , $y = {log}_{3} (x)$ is reflected in the $x$ -axis.
The graph is stretched vertically about the $x$ -axis by a factor of $|a|$ .
If $b < 0$ , the graph is reflected in the $y$ -axis.
The graph is stretched horizontally about the $y$ -axis by a factor of $\frac{1}{|b|}$ .
The graph is translated horizontally $|h|$ unit(s). If $h < 0$ it is shifted left and if $h > 0$ it is shifted right.
The graph is translated vertically $|k|$ unit(s). If $k < 0$ it is shifted down and if $k > 0$ it is shifted up.

Paused Finished

Slide /

A function $y = {log}_{5} (x)$ has been transformed as follows:

Reflected in the $x$ -axis
Reflected in the $y$ -axis
Stretched vertically about the $x$ -axis by a factor of $\frac{1}{3}$
Translated horizontally $10$ units to the left and vertically $8$ units upward

The equation of the transformed function is:

y =

Notes:	1. Use brackets for your logarithm argument. For example $\log_5{(x)}$ is correct, $\log_5{x}$ is not.
	2. You can enter subscripts by first typing an underscore, then the subscript, then right-arrow.

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Möbius Mathematics Readiness

Slide Notes

Glossary

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Navigation

General Observations

Observations on the Value of $c$

Check Your Understanding D

Transformations of Logarithmic Functions

Check Your Understanding E

Möbius Mathematics Readiness

Slide Notes

Glossary

All Slides

Navigation

General Observations

Observations on the Value of c

Check Your Understanding D

Transformations of Logarithmic Functions

Check Your Understanding E

Observations on the Value of $c$