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Graphs of logarithmic functions

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Engineering Maths B, Logarithms
August 25th, 2008 by Craig Rose
Permalink: http://www.craigsmaths.com/ea003-engineering-mathematics-b/graphs-of-logarithmic-functions/

Engineering Maths B, Logarithms
August 25th, 2008 by Craig Rose
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By comparing logarithmic functions with the simple form it is easy to sketch their graphs using a transformational approach.

The simple logarithmic function

First take a look at the simplest logarithmic function and it’s graph:

$f\left(x\right)=\log{_e}\left(x\right)$ where $e>0$ and $e\neq1$ .

Simple log function

The key points to note about this function and sketch are:

There is a vertical asymptote at $x=0$
The x-intercept is at $x=1$
$f\left(e\right)=1$ . That is $\log{_e}e=1$
$f\left(x\right)$ for $0<x<1$ is negative
$f\left(x\right)$ for $x>1$ is positive
$x>0$

A further feature to note is that $f\left(x\right)=\log\left(x\right)$ is the inverse function of $f^{-1}\left(x\right)=e^{x}$ and thus a refection in $y=x$ . This can be applied to all the logarithmic functions as an alternative way to plot them.

The logarithm is the inverse of the exponent

For the purposes of comparison we take $e$ to be Euler’s number. This is useful for many real world applications. An alternative notation for $\log{_e}\left(x\right)$ is $ln\left(x\right)$ .

The general logarithmic functions

A general form can be expressed as:

$f\left(x\right)=a\log{_e}\left(bx-c\right)-d$

Comparing this with the simple case we can observe the various transformations resulting.

Dilation in the y-direction ( $a$ )

$a$ is a dilation (stretched or compressed) in the y-direction. For $0<\left|a\right|<1$ the graph will be stretched and for $\left|a\right|<1$ it will compress. If $a<0$ then the graph will dilate and be reflected in the x-axis. The x-intercept will always be at $\left(1,0\right)$ and the asymptote at $x=0$ .

Dilation in the positive y-direction

Dilation in the y-direction and reflection in x-axis

Dilation in the x-direction ( $b$ )

$b$ is a dilation (stretched or compressed) in the x-direction. For $0<\left|b\right|<1$ the graph will be stretched and for $\left|b\right|<1$ it will compress. If $b<0$ then the graph will dilate and be reflected in the y-axis. The x-intercept will be at the point $\left(\frac{1}{b},0\right)$ .

Dilation in the positive x-direction

Dilation in x-direction with reflection

Translation in the x-direction ( $c$ )

When $c>0$ the entire graph will translate $\left|c\right|$ units to the right. When $c<0$ the entire graph will translate $\left|c\right|$ units to the left. The asymptote then becomes $x=c$ and the x-intercept is $\left(1+c,0\right)$ .

Translation in x-direction

Translation in the y-direction ( $d$ )

When $d>0$ the entire graph will translate $\left|d\right|$ units up. When $d<0$ the entire graph will translate $\left|d\right|$ units down. The asymptote remains at $x=0$ and the x-intercept is $\left(a^{d},0\right)$ .

Translation in y-direction

Sketching the graph

Find the x-intercept

There will always be an x-intercept. This is found by solving $0=a\log{_e}\left(bx-c\right)-d$ for $x$ . When solving this remember that $e^{y}=x$ is another form of $y=log{_e}x$ .

Is there y-intercept?

Substituting $x=0$ there is only a real solution for $y=a\log{_e}\left(-c\right)-d$ when $c<0$ .

Find the y-intercept?

Calculate $y$ for $y=a\log{_e}\left(-c\right)-d$ .

Find the asymptote

The asymptote will be the vertical line $x=c$ .

Apply any reflections

Recall that $a<0$ will reflect the simple logarithmic function in the x-axis and $b<0$ is the reflection in the y-axis.

Sketch the graph

Now you have enough information to do the sketch. To give a better concept of the shape of the graph you may want to sketch in some calculated points as well. One at $x=e$ (the base) would be useful.

Sketch the asymptote as a dotted line.

Tags: graphs, Logarithms, maths tutor hobart

Monday, August 25th, 2008 at 9:03 am
Category: Engineering Maths B, Logarithms.
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anon Says:
November 4th, 2008 at 5:15 pm

thanks for the help
just wondering, what will graphs look like with different bases

e.g. log( base 2) x compared to log (base 3)x and log (base 4) x

Craig Rose Says:
November 5th, 2008 at 7:51 pm

The key feature that identifies the base from a sketch is the point where $y=1$ . For the simple log function (no transformations) this will be at $x=the base$ . As the log of the base is equal to 1. In other words $base^1=base$ .

Graphs of logarithmic functions

The simple logarithmic function

The general logarithmic functions

Sketching the graph

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