Essay on Logarithms: Logarithm and Exponential Functions

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s m h t i r a g Lo rtz a w h c S
Sarah

What is a Logarithm?
Talk amongst yourselves about what you think a logarithm is – have you ever heard this term before? What does the picture above remind you of?

Introduction
True or false: this sea shell shows a exponential growth in the spiral from the inside out.

True!

Exponential Functions
Review: a function that shows a exponential growth or decay
General form: F(x)=ax
There is a horizontal asymptote.

If you were to inverse an exponential function, what do you think it would look like?
This is called a LOGARITHM

John Napier invented the logarithm in 1614.

History of
Logarithms

Scottish mathematician
Picture on left

Napier’s work involved long calculations and large numbers – he wished for a simpler way to carry out his work How did he do it?
Napier created exponential form (ex. 23=8, 24=16, 25=32)
From this, he made his first logarithmic table

History Continued
Henry Briggs and John
Napier collaborated and created the official logarithm form
General form:

y=logbx  by=x Understanding the
Formula
A logarithm is a cause of an effect
An input for a output
A logarithm will be equal to how many of one number do we multiply to get another number?
Ex. How many times do I need to multiply 3 to get 81?
Answer: 3 x 3 x 3 x 3 =81, therefore you need 4 3’s to get
81
The logarithm is 4.

How is it written?
33 xx 33 xx 33 xx 33 =
= 81
81

33 =81
=81
44

Log
Log33 (81)
(81) =
= 44
The
The number number of of 3’s
3’s
you need to get 81 you need to get 81 is is 4.
4.

A logarithmic function uses the word ‘log’ to signify ‘inverse’
It does not mean to click the log button on your calculator! You can only use this if the base is
10.

The base of the function is 3.
In general form, the equation is logbx=y Helpful Tips
If there is no base, it is assumed that b=10.
Ex. What is the logarithm of logx(1000)?
10 to the power of what equals 1000? 103
Logx(1000)=3

If you are confused, it helps to say it out loud.
Remember that logarithms always answer this question:
How many of my base do I need to get my y?

y=logbx by=x Switching Between Exponential and Logarithmic
On the next page are logarithmic and exponential functions. For each one, switch to the opposite function. Try to improve your time as you go through the questions.

Questions!
1. Switch the following exponential functions into logarithmic functions: i. 16=42 ii. 10000=104 iii. 125=53
2. Switch the following logarithmic functions to exponential functions: i. Log3(27)=3 ii. Log(100)=2 iii. Log4(256)=4

Answers:
1.
i. log4(16)=2 ii. Log(10000)=4 iii. Log5=3
2.
i. 27=33 ii. 100=102 iii. 256=44

Practise answering these logarithms! Questions

Answers

1. Log5(25)

1. Log5(25)=2

2. Logb(49)=2

2. b=7

3. Log(x)=5

3. x=100000

4. Log3(243)

4. Log3(243)=5

5. Log1/2(1/10)

5. Log1/2(1/10)=5

Transformations!
Like other functions, there are transformations that can be performed to change the logarithmic function. We will now explore them.

Horizontal
Translations

A horizontal translation is performed by adding or subtracting a value to the x value. You have already learned about these types of transformations – they work
the
Show More

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