All About LogarithmsAll About Logarithms

A Practical and Most Useful A Practical and Most Useful

Guide to LogarithmsGuide to Logarithms

byby

Mr. HansenMr. Hansen

John NapierJohn 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 In computing tables, these large numbers may again be made still larger by placing a period after the number and adding still larger by placing a period after the number and adding ciphers. ... In numbers distinguished thus by a period in their ciphers. ... In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the midst, whatever is written after the period is a fraction, the denominator of which is unity with as many ciphers after it as denominator of which is unity with as many ciphers after it as there are figures after the period.” 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!

Johannes Kepler (1571-1630)

A typical page from one of Kepler’s notebooks

Napier’s BonesNapier’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 AppearLogarithms 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 WorkHow Logarithms Work

Logarithms are based on exponential functions.Logarithms are based on exponential functions.

Common logs are based on ten raised to a power: Common logs are based on ten raised to a power: x = 10x = 10yy

Natural logs, which are based on the number e Natural logs, which are based on the number e

raised to araised to apower, are used mostly in higher and theoreticalpower, are used mostly in higher and theoreticalmathematics:mathematics:

x = ex = eyy

Either of these functions can be graphed in the Either of these functions can be graphed in the normal way and produce the typical exponential normal way and produce the typical exponential curve. Notice how x and y are interchanged in curve. Notice how x and y are interchanged in these expressions.these expressions.

Logarithmic NotationLogarithmic Notation

For logarithmic functions we use the notationFor logarithmic functions we use the notation::

loglogaa(x) or log(x) or logaaxx

This is read “log, base a, of x.” ThusThis is read “log, base a, of x.” Thus,,

y = logy = logaaxx meansmeans x = ax = ayy

And so a logarithm is simply an exponent of some And so a logarithm is simply an exponent of some basebase..

Inverse Relations and FunctionsInverse 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 FAQsLogarithmic Function FAQs

Logarithms are a mathematical tool originally Logarithms are a mathematical tool originally invented to reduce arithmetic computations.invented to reduce arithmetic computations.

Multiplication and division are reduced to simple Multiplication and division are reduced to simple addition and subtraction.addition and subtraction.

Exponentiation and root operations are reduced Exponentiation and root operations are reduced more simple exponent multiplication or division.more simple exponent multiplication or division.

Changing the base of numbers is simplified.Changing the base of numbers is simplified.

Scientific and graphing calculators provide Scientific and graphing calculators provide logarithm functions for base 10 (common) and logarithm functions for base 10 (common) and base e (natural) logs. Both log types can be used base e (natural) logs. Both log types can be used for ordinary calculations.for ordinary calculations.

Exponential & Logarithmic FunctionsExponential & Logarithmic Functions

Exponential functions always have the variable in Exponential functions always have the variable in the exponent:the exponent:

f(x) = 2f(x) = 2xx is an exponential function is an exponential function

f(x) = xf(x) = x22 is is notnot an exponential function an exponential function

Definition: The function Definition: The function f(x) = af(x) = axx, where a is a , where a is a positive number constant other than 1, is called positive number constant other than 1, is called an an exponential function, base a.exponential function, base a.

Graphing Exponential Functions and Graphing Exponential Functions and LogsLogs

Let’s graph Let’s graph y = logy = log33xx::

Observation: because x and y Observation: because x and y can be interchanged in can be interchanged in this equation, the graph of this equation, the graph of x = 3yx = 3y is a reflection of is a reflection of y y = 3x= 3x across the line across the line y = xy = 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 LogarithmsLaws of Exponents and Logarithms

Because logarithms are exponents, the laws of Because logarithms are exponents, the laws of exponents apply toexponents apply to

all logarithmic operations. These laws include:all logarithmic operations. These laws include:

Law for MultiplicationLaw for Multiplicationbbxx ∙ b ∙ byy = b = bx + yx + y

Law for DivisionLaw for Divisionbbxx / b / byy = b = bx – yx – y

Law for Power of a PowerLaw for Power of a Power(b(bxx) ) y = y = bbx yx y

Law for Negative ExponentsLaw for Negative Exponents bb-x-x = 1 / b = 1 / bxx

Exponential and Logarithmic Exponential and Logarithmic RelationshipsRelationships

Conversion between log and exponential forms is Conversion between log and exponential forms is often aoften a

convenient way to solve problems.convenient way to solve problems.

Because x = aBecause x = ayy and y = log a and y = log ax x are equivalent, then:are equivalent, then:22xx = 8 is the same as x = log = 8 is the same as x = log2288

Log problems are solved the same way:Log problems are solved the same way:loglog22x = -3, the equivalent of 2x = -3, the equivalent of 2-3-3

And by the laws of exponents, we obtain:And by the laws of exponents, we obtain: x = 1/2x = 1/233 or x = 1 / 8 or x = 1 / 8

Exponents: Characteristic & Exponents: Characteristic & MantissaMantissa

The exponent of a number The exponent of a number NN consists of an integer consists of an integer or characteristic and a mantissa that follows the or characteristic and a mantissa that follows the decimal point. decimal point.

The characteristic is determined by the number of The characteristic is determined by the number of places the decimal point is moved from its position places the decimal point is moved from its position when when NN is written in scientific notation. is written in scientific notation.

The The mantissamantissa of an exponent is a non-ending of an exponent is a non-ending decimal fraction following the characteristic. This decimal fraction following the characteristic. This number is most often found in a table of number is most often found in a table of logarithms.logarithms.

Tables of logarithms usually give mantissa values Tables of logarithms usually give mantissa values in 4 or 5 decimal places. The user must manually in 4 or 5 decimal places. The user must manually calculate the characteristic.calculate the characteristic.

Properties of Logarithmic FunctionsProperties of Logarithmic FunctionsWe always assume that a is positive (<> 1), and is a We always assume that a is positive (<> 1), and is a

constant soconstant soit can serve as a logarithm base. This is a proof of the it can serve as a logarithm base. This is a proof of the

firstfirsttheorem of logarithms:theorem of logarithms:

Proof: For any positive numbers x & y, logProof: For any positive numbers x & y, logaa (xy) = log (xy) = logaa x x + log+ logaa y y

Let b = logLet b = logaa x and c = log x and c = logaa y. This is equivalent to: y. This is equivalent to:x = ax = abb andand y = a y = acc

Now multiply x and y:Now multiply x and y:xy = axy = abbaacc, , or, by a law of exponentsor, by a law of exponents, = a, = abb + c+ c

As a logarithmic statement we can now write:As a logarithmic statement we can now write:

loglogaa(xy) = b + c(xy) = b + c

And replacing the values of b and c, our solution is:And replacing the values of b and c, our solution is:

loglogaa (xy) = log (xy) = logaa x + log x + logaa y y

Additional Log ComputationsAdditional Log Computations

Other log theorems show these relationships:Other log theorems show these relationships:

loglogaaxxpp = p ∙ log = p ∙ logaa x x - raise a number to a power by - raise a number to a power by multiplying the log of the multiplying the log of the

number; number; take the root of a number take the root of a number by dividingby dividing

the log of that number.the log of that number.

logloga a x / y = logx / y = logaa x x - log - logaa y - divide two numbers by y - divide two numbers by

subtracting the logs subtracting the logs of the twoof the two

numbers.numbers.

loglogbb N = log N = logaa N / log N / logaa b b - change the base of - change the base of number N by number N by dividing its log by the dividing its log by the log of the new log of the new base.base.

Using LogarithmsUsing Logarithms Logarithm calculations produce answers as an Logarithm calculations produce answers as an

exponent. exponent.

To find the actual numeric solution of the To find the actual numeric solution of the calculation, the “antilogarithm” of the result must calculation, the “antilogarithm” of the result must be found.be found.

When using a calculator, this is done by raising the When using a calculator, this is done by raising the base number to the power of that exponent.base number to the power of that exponent.

Example: Using common logs, find the value of 37Example: Using common logs, find the value of 3733..

Step 1: Using a calculator, find the common Step 1: Using a calculator, find the common logarithm of 37 (1.58201724), then multiply logarithm of 37 (1.58201724), then multiply

that by 3, that by 3, (4.704605172).(4.704605172).

Step 2: Use a calculator to find the value ofStep 2: Use a calculator to find the value of

10^4.704605172. The answer is 50653.10^4.704605172. The answer is 50653.

Log Calculations Using a Log TableLog Calculations Using a Log Table

The Problem: multiply 37 by 143 using the The Problem: multiply 37 by 143 using the handout log tables. This table, originally produced handout log tables. This table, originally produced in 1939, is accurate to four decimal places.in 1939, is accurate to four decimal places.

Determine the characteristic of these numbers:Determine the characteristic of these numbers:

37 = 3.7 ∙ 1037 = 3.7 ∙ 1011; the characteristic is 1.; the characteristic is 1.

143 = 1.43 ∙ 10143 = 1.43 ∙ 1022 ; the characteristic is ; the characteristic is 2.2.

Using the table, find the number 37 in the left-Using the table, find the number 37 in the left-most column, and read its value in the second (0 most column, and read its value in the second (0 column) as .5051. The log of 37 is thus 1.5682. column) as .5051. The log of 37 is thus 1.5682.

You can check this with your calculator by You can check this with your calculator by computing 10computing 101.56821.5682 which is 36.99985312. which is 36.99985312.

Next find the log of 143. For this mantissa, we Next find the log of 143. For this mantissa, we use the log of 14.3; look this up under the use the log of 14.3; look this up under the number 14, then go to column 3. This mantissa is number 14, then go to column 3. This mantissa is .1553. (Notice how the first digit of the mantissa .1553. (Notice how the first digit of the mantissa is only printed in column 0.) The characteristic is is only printed in column 0.) The characteristic is now added to the mantissa, making the log of now added to the mantissa, making the log of 143 to be 2.1553.143 to be 2.1553.

Now add the two logs together: 1.582 + 2.1553 Now add the two logs together: 1.582 + 2.1553 = 3.7235. This is the logarithm of our answer.= 3.7235. This is the logarithm of our answer.

The answer, (10The answer, (103.72353.7235 or 5291 by calculator), is or 5291 by calculator), is discovered by finding the number from the table discovered by finding the number from the table that is closest to .7235, our mantissa, then that is closest to .7235, our mantissa, then multiplying it by 10 raised to the power of the multiplying it by 10 raised to the power of the characteristic.characteristic.

Our mantissa is found in column 9 under the Our mantissa is found in column 9 under the number 52. The resulting number, then, is 5.29, number 52. The resulting number, then, is 5.29, remembering that our mantissa is less than 1. remembering that our mantissa is less than 1.

The final answer is calculated as 5.29 times 10 The final answer is calculated as 5.29 times 10 raised to the power of the characteristic 3, or raised to the power of the characteristic 3, or 1000. The result, 5290, is one less than the 1000. The result, 5290, is one less than the actual answer of 5291, but is within 99.98% of actual answer of 5291, but is within 99.98% of the actual value.the actual value.

We will now repeat this multiplication, this time We will now repeat this multiplication, this time using the log values produced by our calculator using the log values produced by our calculator rather than the tables.rather than the tables.

Log Computations Using a Log Computations Using a CalculatorCalculator

The four-place log tables provide limited accuracy The four-place log tables provide limited accuracy compared tocompared to

that offered by today’s scientific and graphic calculators.that offered by today’s scientific and graphic calculators.

Henry Briggs (1560-1631) produced the most accurate Henry Briggs (1560-1631) produced the most accurate table of common logs for more than 300 years when in table of common logs for more than 300 years when in 1631 he published a book of 30,000 logarithms 1631 he published a book of 30,000 logarithms accurate to 14 decimal places.accurate to 14 decimal places.

In 1952 Professor Alexander J. Thompson published his In 1952 Professor Alexander J. Thompson published his 20-figure log tables. This project was started in 1924 to 20-figure log tables. This project was started in 1924 to celebrate the tercentenary of Brigg’s tables, but when celebrate the tercentenary of Brigg’s tables, but when finished, the tables could have been generated in a finished, the tables could have been generated in a matter of minutes by newly developed computers.matter of minutes by newly developed computers.

The following exercise repeats the multiplication just The following exercise repeats the multiplication just done, this time using a graphing calculator that carries done, this time using a graphing calculator that carries results to 14 decimal places with 2 digit exponents.results to 14 decimal places with 2 digit exponents.

On your calculator, start by pressing the On your calculator, start by pressing the LOGLOG button, then the number button, then the number 3737, followed by the closing , followed by the closing parenthesis. Press parenthesis. Press ENTERENTER and observe the results: and observe the results:

LOG(37) = 1.568201724LOG(37) = 1.568201724

Now do the same, this time with the number Now do the same, this time with the number 143143..

LOG(143) = 2.155336037LOG(143) = 2.155336037

Press the Press the 22ndnd button followed by the button followed by the 1010xx (LOG) (LOG) button, then enter the sum of these two numbers button, then enter the sum of these two numbers (3.723537761).(3.723537761).

The result, 5290.999994, is much closer to the The result, 5290.999994, is much closer to the actual product of 37 and 143 (5291) than we actual product of 37 and 143 (5291) than we obtained using the four-place log tables. This result obtained using the four-place log tables. This result is within 99.99999998% of the actual value.is within 99.99999998% of the actual value.

ConclusionConclusion Logarithms were originally devised to simplify Logarithms were originally devised to simplify

arithmetic.arithmetic.

Logarithms are simply exponents of some base (usually Logarithms are simply exponents of some base (usually base 10), and therefore follow all the rules of exponents.base 10), and therefore follow all the rules of exponents.

Logarithms can use any base, but only two bases are Logarithms can use any base, but only two bases are normally used today: Common Logs use a base of 10, normally used today: Common Logs use a base of 10, and Natural Logs use e as their base.and Natural Logs use e as their base.

The modern calculator has nearly eliminated the need The modern calculator has nearly eliminated the need or use of logarithms in ordinary calculations. or use of logarithms in ordinary calculations. Theoretical math and physics, however, frequently Theoretical math and physics, however, frequently employ the use of Natural Logs.employ the use of Natural Logs.