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, later) 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 = 102
In 102, 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).
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.)
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Exponential notation for even (whole number) powers of 10.
Exponential
Number
Notation
Factors of 10
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1
10
100
1,000
10,000
100,000
1,000,000
100
101
102
103
104
105
106
(any number raised to the zero power = 1)
10 x 1 = 10 (‘times 1’ is implied in all of these)
10 x 10 = 100
10 x 10 x 10 = 1,000
10 x 10 x 10 x 10 = 10,000
10 x 10 x 10 x 10 x 10 = 100,000
10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
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Whole-number powers of 10 are simple: The exponent is
the number of zeroes used when the number is written in
ordinary notation. 100 = 102, 10,000 = 104, 1,000,000 = 106 ,
1,000,000,000 = 109, …
What do you do with positive numbers that are
smaller than 1, like 0.01?
The convention is pretty simple:
0.01 = 1/100 = 1/102 = 10-2
To go at it from the other direction:
10-2 = 1/102 = 1/100 = 0.01
How do you write 0.001 in exponential notation?
0.001 = 1/1,000 = 1/103 = 10-3
How about 0.1?
0.1 = 1/10 = 1/101 = 10-1
(There’s an easy zero-counting system for these too. Soon.)
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Exponential notation for positive numbers less than 1.
Exponential
Number
Notation
Factors of 10
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0.1
10-1
1/10 = 1/101
0.01
10-2
1/(10x10) = 1/102
0.001
10-3
1/(10x10x10) = 1/103
0.0001
10-4
1/(10x10x10x10) = 1/104
0.00001
10-5
1/(10x10x10x10x10) = 1/105
0.000001
10-6
1/(10x10x10x10x10x10) = 1/106
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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.
Last point on negative exponents: Don’t
lose track of the basic idea, which is pretty
straightforward:
10-1 = 1/101 = 1/10 = 0.1
10-2 = 1/102 = 1/100 = 0.01
10-3 = 1/103 = 1/1,000 = 0.001
10-4 = 1/104 = 1/10,000 = 0.0001
.
.
.
That is pretty much it.
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.
base
2
10
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.
If an exponent is not a scary idea, then a
logarithm cannot be scary. They are
different names for the exact same thing.
Q: What is the base-10 logarithm (log10) of 100?
A: Write 100 in exponential notation. The exponent
is the log10.
100 = 102, so log10(100) = 2
That 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)
log10(0.01) = -2 (because 0.01 = 10-2)
log10(0.000001) = -6 (because 10 = 10-6)
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, …).
log10(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 log10 of 4, right?
log10(10,000) = 4
log10(100,000) = 5
So, log10(20,000) must be more than 4 but less
than 5.
But what is it exactly?
Three ways to find the answer:
1. Calculate it using Isaac Newton’s method.
2. Look it up in a table of logarithms. (That is
actually more confusing than Newton’s method.)
3. 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)
Now, the very last thing: Arithmetic using
exponents. You may remember these from HS.
Multiplication:
102 x 104 = 102+4 = 106 (add the exponents)
105 x 103 = 105+3 = 108 (ditto)
Division:
105 / 102 = 105-2 = 103 (subtract the exponents)
103 / 105 = 103-5 = 10-2 (ditto)
Exponentiation (raising a number in exp. notation to a power):
(103)2 = 103x2 = 106 (multiply the exponents)
(105)3 = 105x3 = 1015 (multiply the exponents)
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.
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.