This course requires very little math just a small number of fairly simple formulas One math concept well need for the decibel scale later is exponential notation Its not hard and youve already had it ID: 278327
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Slide1
Exponential Notation
This course requires very little math – just a small number of fairly simple formulas. One math concept we’ll need (for the decibel
scale - soon)
is
exponential notation.
It’s not hard, and you’ve already had it.
Exponential notation provides a convenient way to represent very large or very small numbers. Simple example:
100 = 10 x 10 = 10
2
In 10
2
, 10 is the
base,
and
2 is the
exponent.
The exponent gives the number of times the base is used as a factor (i.e., used in multiplying the base by itself).Slide2
Note
:
The base does not need to be 10
(although that’s the only one we’ll need).
It can be 2
(the base that’s used in digital computers – i.e., binary, 0s and 1s)
; it can be 16
(hexadecimal, a coding scheme used by computer nerds)
; it can be anything it wants to be.
The decibel scale uses base 10, so that’s all we’ll work with.
(We will not get to this for a while.)Slide3
___________________________________________________________________________
___________________________________________________________________________
Exponential
notation for even
(whole number) powers
of 10.
Exponential
Number Notation Factors of 10
___________________________________________________________________________
1 10
0
(any number raised to the zero power = 1)
10 10
1
10 x 1 =
10 (‘times 1’ is implied in all of these)
100 10
2
10 x 10 = 100
1,000 10
3
10 x 10 x 10 = 1,000
10,000 10
4
10 x 10 x 10 x 10 = 10,000
100,000 10
5
10 x 10 x 10 x 10 x 10 = 100,000
1,000,000 10
6
10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
___________________________________________________________________________
___________________________________________________________________________
Whole-number powers of 10 are simple:
The exponent is the number of zeroes used when the number is written in ordinary notation.
100 = 10
2
, 10,000 = 10
4
, 1,000,000 = 10
6
, 1,000,000,000 = 10
9
, …Slide4
What do you do with
positive numbers that are smaller than 1,
like 0.01?
The convention is pretty simple:
0.01 = 1/100 = 1/10
2
= 10
-2
To go at it from the other direction:
10
-2 =
1/10
2
= 1/100 = 0.01
H
ow do you write 0.001 in exponential notation?
0.001 = 1/1,000 = 1/10
3
= 10
-3
How about 0.1?
0.1 = 1/10 = 1/10
1
= 10
-1
(There’s an easy zero-counting system for these too. Soon.)Slide5
____________________________________________________________________
____________________________________________________________________
Exponential
notation for
positive numbers
less than 1.
Exponential
Number
Notation
Factors of 10
____________________________________________________________________
0.1 10
-1
1/10
= 1/10
1
0.01 10
-2
1
/(10x10) = 1/10
2
0.001 10
-3
1
/(10x10x10) = 1/10
3
0.0001
10
-4
1
/(10x10x10x10) = 1/10
4
0.00001
10
-5
1
/(10x10x10x10x10) = 1/10
5
0.000001 10
-6
1
/(10x10x10x10x10x10) = 1/10
6
____________________________________________________________________
____________________________________________________________________
Zero-counting rule for whole-number powers of 10: The exponent is the
number of zeroes
plus 1
(but with a negative sign);
e.g. 0.01, 1 zero+1, exponent=-2; 0.0001, 3 zeros+1, exponent=-4; 0.000001, 5 zeros+1, exponent=-6. Slide6
Last point on negative exponents:
Don’t lose track of the basic idea, which is pretty straightforward:
10
-1
= 1/10
1
= 1/10 = 0.1
10
-2
= 1/10
2
= 1/100 = 0.01
10
-3
= 1/10
3
= 1/1,000 = 0.001
10
-4
= 1/10
4
= 1/10,000 = 0.0001
.
.
.
That is pretty much it.Slide7
The Dreaded Logarithm
We’re going through all this because we will need it later when we talk about the decibel (dB) scale, used (mainly) for the measurement of sound intensity.
The dB scale is logarithmic.
Most students are comfortable with the concept of an exponent, but logarithms sometimes strike fear. But this is usually because the concept is not taught well.
Here’s the main joke:
a logarithm
is
an exponent.
A log is not like an exponent, it
is
an exponent.Slide8
10
2
base
exponent
OR
power
OR
logarithm
All three terms –
exponent,
power,
and
logarithm
– are
completely interchangeable.
An exponent is a power, a power is a logarithm,
a logarithm is an exponent
.
I
f an exponent is not a scary idea, then a logarithm cannot be scary.
They are different names for the exact same thing.Slide9
Q: What is the base-10 logarithm
(log
10
)
of 100?
A: Write 100 in exponential notation.
The exponent
is
the log10. 100 = 102, so log10(100) = 2That is the whole thing
. log10
(1,000) = 3 (because 1,000 = 103) log10(10) = 1 (because 10 = 101) log10(100,000) = 5 (because 100,000 = 105) log10(1,000,000) = 6
(because 1,000,000 = 106)
log
10
(0.01) =
-2
(because 0.01 = 10
-2
)
log
10
(0.000001) =
-6
(because 10 = 10
-6
)Slide10
Ok, we know how to find logs for numbers that are whole-number powers of 10 (1, 10, 100, 1,000, 0.1, 0.01, …).
log
10
(20,000) = ?
We know we can’t just count zeroes because 10,000 has 4 zeroes, and 20,000 also has 4 zeroes, and 30,000 also has 4 zeroes, ... They can’t all have a log
10
of 4, right?
log
10(10,000) = 4 log10(100,000) = 5So, log10(20,000) must be more than 4 but less than 5.
But what is it exactly?Slide11
Three ways to find the answer:
Calculate it using Isaac Newton’s method.
Look it up in a table of logarithms.
(That is actually more confusing than Newton’s method.)
Type the number into a calculator and press the “log” button.
I recommend #3. You’ll need a calculator with a log button.
You want the button that says “log”, not the button that says “
ln
”. (The ‘ln’ key also calculates a logarithm, but a different kind.)
So, what is the log10(20,000)?
(A: ~4.301)Slide12
Now, the very last thing: Arithmetic using exponents. You may remember these from HS.
Multiplication:
10
2
x 10
4
= 10
2+4 = 10
6
(add the exponents) 105 x 103 = 105+3 = 108 (ditto)
Division:
10
5
/ 10
2
= 10
5-2
= 10
3
(subtract the exponents)
10
3
/ 10
5
= 10
3-5
= 10
-2
(ditto)
Exponentiation
(raising a number in exp. notation to a power)
:
(10
3
)
2
= 10
3x2
= 10
6
(multiply the exponents)
(10
5
)
3
= 10
5x3
= 10
15
(multiply the exponents)Slide13
We are going to need this last one especially:
Squaring a number in exponential notation is the same as multiplying the exponent (
also known as the log
) by 2.
If you understand this rule, something that is apparently obscure about the dB formula will make sense. If you don’t understand it, then you just have to memorize this thing,
which is the hard way.Slide14
THE BIG IDEA
If you understand what an exponent is
(and all of you do),
then you automatically understand what a logarithm is.
A log is nothing more than an exponent with a name that can be intimidating.