Transcript The Slide Rule

All About Logarithms
A Practical and Most Useful
Guide to Logarithms
by
Mr. Hansen
John Napier
John Napier, a 16th Century
Scottish scholar, contributed
a host of mathematical
discoveries.
John Napier (1550 – 1617)
He is credited with creating the first
computing machine, logarithms and
was the first to describe the
systematic use of the decimal point.
Other contributions include a
mnemonic for formulas used in
solving spherical triangles and two
formulas known as Napier's
analogies.
“In computing tables, these large numbers may again be made still larger
by placing a period after the number and adding ciphers. ... In numbers
distinguished thus by a period in their midst, whatever is written after
the period is a fraction, the denominator of which is unity with as many
ciphers after it as there are figures after the period.”
Napier lived during a time when
revolutionary astronomical
discoveries were being made.
Copernicus’ theory of the solar
system was published in 1543, and
soon astronomers were calculating
planetary positions using his
ideas.
But 16th century arithmetic was
barely up to the task and Napier
became interested in this problem.
Nicolaus Copernicus (1473-1543)
Even the most basic astronomical
arithmetic calculations are ponderous.
Johannes Kepler (1571-1630) filled nearly
1000 large pages with dense arithmetic
while discovering his laws of planetary
motion!
A typical page from one of
Kepler’s notebooks
Johannes Kepler (1571-1630)
Napier’s Bones
In 1617, the last year of his life,
Napier invented a tool called
“Napier's Bones” which reduces the
effort it takes to multiply numbers.
“Seeing there is nothing that is so troublesome to
mathematical practice, nor that doth more molest and
hinder calculators, than the multiplications, divisions...
I began therefore to consider in my mind by what
certain and ready art I might remove those hindrances.”
Logarithms Appear
The first definition of the logarithm was constructed by Napier and
popularized by a pamphlet published in 1614, two years before his
death. His goal: reduce multiplication, division, and root extraction
to simple addition and subtraction.
Napier defined the "logarithm" L of a number N by:
N==10^7(1-10^(-7))^L
This is written as NapLog(N) = L or NL(N) = L
This definition leads to these remarkable relations
sqrt(N_1 N_2)
=
10^7(1-10^(-7))^((L_1+L_2)/2)
10^(-7)N_1 N_2
=
10^7(1-10^(-7))^(L_1+L_2)
10^7(N_1) / (N_2)
=
10^7(1-10^(-7))^(L_1-L_2)
which give the identities:
NapLog(sqrt(N_1 N_2))
=
1/2(NapLogN_1+NapLogN_2)
NapLog(10^(-7) N_1N_2)
=
NapLogN_1+NapLogN_2
NapLog(10^7(N_1)/(N_2))
=
NapLogN_1-NapLogN_2
While Napier's definition for logarithms is different from the
modern one, it transforms multiplication and division into
addition and subtraction in exactly the same way.
How Logarithms Work
Logarithms are based on exponential functions.

Common logs are based on ten raised to a power:
x = 10y

Natural logs, which are based on the number e raised to a
power, are used mostly in higher and theoretical
mathematics:
x = ey

Either of these functions can be graphed in the normal
way and produce the typical exponential curve. Notice
how x and y are interchanged in these expressions.
Logarithmic Notation

For logarithmic functions we use the notation:
loga(x) or logax

This is read “log, base a, of x.” Thus,
y = logax means x = ay

And so a logarithm is simply an exponent of some base.
Inverse Relations and Functions
We show an inverse
function using the notation
f(x)-1.
• A function is inverted by
interchanging the x and y
values, then resolving for y.
• Inverting a function
reflects it across the line
x = y.
Logarithmic Function FAQs

Logarithms are a mathematical tool originally invented
to reduce arithmetic computations.

Multiplication and division are reduced to simple
addition and subtraction.

Exponentiation and root operations are reduced more
simple exponent multiplication or division.

Changing the base of numbers is simplified.

Scientific and graphing calculators provide logarithm
functions for base 10 (common) and base e (natural) logs.
Both log types can be used for ordinary calculations.
Exponential & Logarithmic Functions

Exponential functions always have the variable in the
exponent:
f(x) = 2x is an exponential function
f(x) = x2 is not an exponential function

Definition: The function f(x) = ax, where a is a positive
number constant other than 1, is called an exponential
function, base a.
Graphing Exponential Functions and Logs
Let’s graph y = log3x:
Observation: because x and y can
be interchanged in this
equation, the graph of x = 3y is
a reflection of y = 3x across the
line y = x.
Since a0 (a<>0) = 1, the graph of y
= logax, for any a, has the intercept
(1,0).
As can be seen from this function,
the domain of x is all positive real
numbers, and the range is all real
numbers.
Laws of Exponents and Logarithms
Because logarithms are exponents, the laws of exponents apply to
all logarithmic operations. These laws include:




Law for Multiplication
bx ∙ by = bx + y
Law for Division
bx / b y = bx – y
Law for Power of a Power
(bx) y = bx y
Law for Negative Exponents
b-x = 1 / bx
Exponential and Logarithmic
Relationships
Conversion between log and exponential forms is often a
convenient way to solve problems.
Because x = ay and y = log ax are equivalent, then:
2x = 8 is the same as x = log28
Log problems are solved the same way:
log2x = -3, the equivalent of 2-3
And by the laws of exponents, we obtain:
x = 1/23 or x = 1 / 8
Exponents: Characteristic & Mantissa

The exponent of a number N consists of an integer or
characteristic and a mantissa that follows the decimal
point.

The characteristic is determined by the number of places
the decimal point is moved from its position when N is
written in scientific notation.

The mantissa of an exponent is a non-ending decimal
fraction following the characteristic. This number is most
often found in a table of logarithms.

Tables of logarithms usually give mantissa values in 4 or
5 decimal places. The user must manually calculate the
characteristic.
Properties of Logarithmic Functions
We always assume that a is positive (<> 1), and is a constant so
it can serve as a logarithm base. This is a proof of the first
theorem of logarithms:
Proof: For any positive numbers x & y, loga (xy) = loga x + loga y
Let b = loga x and c = loga y. This is equivalent to:
x = ab and y = ac
Now multiply x and y:
xy = abac, or, by a law of exponents, = ab + c
As a logarithmic statement we can now write:
loga(xy) = b + c
And replacing the values of b and c, our solution is:
loga (xy) = loga x + loga y
Additional Log Computations
Other log theorems show these relationships:
logaxp = p ∙ loga x - raise a number to a power by
multiplying the log of the number;
take the root of a number by dividing
the log of that number.
loga x / y = loga x - loga y - divide two numbers by
subtracting the logs of the two
numbers.
logb N = loga N / loga b - change the base of number N by
dividing its log by the log of the new
base.
Using Logarithms

Logarithm calculations produce answers as an exponent.

To find the actual numeric solution of the calculation, the
“antilogarithm” of the result must be found.

When using a calculator, this is done by raising the base
number to the power of that exponent.

Example: Using common logs, find the value of 373.
Step 1: Using a calculator, find the common
logarithm of 37 (1.58201724), then multiply that by 3,
(4.704605172).
Step 2: Use a calculator to find the value of
10^4.704605172. The answer is 50653.
Log Calculations Using a Log Table

The Problem: multiply 37 by 143 using the handout log
tables. This table, originally produced in 1939, is accurate
to four decimal places.

Determine the characteristic of these numbers:
37 = 3.7 ∙ 101; the characteristic is 1.
143 = 1.43 ∙ 102 ; the characteristic is 2.

Using the table, find the number 37 in the left-most
column, and read its value in the second (0 column) as
.5051. The log of 37 is thus 1.5682.

You can check this with your calculator by computing
101.5682 which is 36.99985312.

Next find the log of 143. For this mantissa, we use the
log of 14.3; look this up under the number 14, then go to
column 3. This mantissa is .1553. (Notice how the first
digit of the mantissa is only printed in column 0.) The
characteristic is now added to the mantissa, making the
log of 143 to be 2.1553.

Now add the two logs together: 1.582 + 2.1553 = 3.7235.
This is the logarithm of our answer.

The answer, (103.7235 or 5291 by calculator), is discovered
by finding the number from the table that is closest to
.7235, our mantissa, then multiplying it by 10 raised to
the power of the characteristic.

Our mantissa is found in column 9 under the number 52.
The resulting number, then, is 5.29, remembering that
our mantissa is less than 1.

The final answer is calculated as 5.29 times 10 raised to
the power of the characteristic 3, or 1000. The result,
5290, is one less than the actual answer of 5291, but is
within 99.98% of the actual value.

We will now repeat this multiplication, this time using
the log values produced by our calculator rather than the
tables.
Log Computations Using a Calculator
The four-place log tables provide limited accuracy compared to
that offered by today’s scientific and graphic calculators.

Henry Briggs (1560-1631) produced the most accurate
table of common logs for more than 300 years when in
1631 he published a book of 30,000 logarithms accurate
to 14 decimal places.

In 1952 Professor Alexander J. Thompson published his
20-figure log tables. This project was started in 1924 to
celebrate the tercentenary of Brigg’s tables, but when
finished, the tables could have been generated in a
matter of minutes by newly developed computers.

The following exercise repeats the multiplication just
done, this time using a graphing calculator that carries
results to 14 decimal places with 2 digit exponents.

On your calculator, start by pressing the LOG button,
then the number 37, followed by the closing parenthesis.
Press ENTER and observe the results:
LOG(37) = 1.568201724

Now do the same, this time with the number 143.
LOG(143) = 2.155336037

Press the 2nd button followed by the 10x (LOG) button,
then enter the sum of these two numbers (3.723537761).

The result, 5290.999994, is much closer to the actual
product of 37 and 143 (5291) than we obtained using the
four-place log tables. This result is within 99.99999998%
of the actual value.
Conclusion

Logarithms were originally devised to simplify
arithmetic.

Logarithms are simply exponents of some base (usually
base 10), and therefore follow all the rules of exponents.

Logarithms can use any base, but only two bases are
normally used today: Common Logs use a base of 10,
and Natural Logs use e as their base.

The modern calculator has nearly eliminated the need
or use of logarithms in ordinary calculations. Theoretical
math and physics, however, frequently employ the use
of Natural Logs.