Logarithmic

Functions

y = logax if and only if x = a y

The logarithmic function to the base a, where a > 0 and a 1 is defined:

2416

exponential form

logarithmic form

Convert to log form: 216log4 Convert to exponential form:

38

1log2 8

12 3

When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.

LOGS = EXPONENTS

With this in mind, we can answer questions about the log:

16log2

This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?)

4

9

1log3

What exponent do you put on the base of 3 to get 1/9? (hint: think negative)2

1log4What exponent do you put on the base of 4 to get 1?0

3log3

When working with logs, re-write any radicals as rational exponents.2

1

3 3logWhat exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)

2

1

Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on what we learned about the graphs of exponentials.

Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa.

Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.

Characteristics about the Graph of an Exponential Function a > 1 xaxf

1. Domain is all real numbers

2. Range is positive real numbers3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

4. The y intercept is always (0,1) because a 0 = 15. The graph is always increasing6. The x-axis (where y = 0) is a horizontal asymptote for x -

Characteristics about the Graph of a Log Function where a > 1 xxf alog

1. Range is all real numbers

2. Domain is positive real numbers

3. There are no y intercepts

4. The x intercept is always (1,0) (x’s and y’s trade places)

5. The graph is always increasing6. The y-axis (where x = 0) is a vertical asymptote

Exponential Graph Logarithmic Graph

Graphs of inverse functions are reflected about the line y = x

Transformation of functions apply to log functions just like they apply to all other functions so let’s try a couple.

xxf 10log

xxf 10log2

up 2

1log10 xxf

left 1

xxf 10logReflect about x axis

Remember our natural base “e”? We can use that base on a log.

7182828.2logeWhat exponent do you put on e to get 2.7182828?

17182828.2log e

Since the log with this base occurs in nature frequently, it is called the natural log and is abbreviated ln.

ln

17182828.2ln

Your calculator knows how to find natural logs. Locate the ln button on your calculator. Notice that it is the same key that has ex above it. The calculator lists functions and inverses using the same key but one of them needing the 2nd (or inv) button.

Another commonly used base is base 10.A log to this base is called a common log.Since it is common, if we don't write in the base on a log it is understood to be base 10.

What exponent do you put on 10 to get 100?

This common log is used for things like the richter scale for earthquakes and decibles for sound.

100log

Your calculator knows how to find common logs. Locate the log button on your calculator. Notice that it is the same key that has 10x above it. Again, the calculator lists functions and inverses using the same key but one of them needing the 2nd (or inv) button.

2

1000

1log 3 What exponent do you put

on 10 to get 1/1000?

The secret to solving log equations is to re-write the log equation in exponential form and then solve.

312log2 x Convert this to exponential form

128 x

x27

x2

7

check:

312

72log2

1223 x

38log2 This is true since 23 = 8

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au