All About Logarithms A Practical and Most Useful Guide to Logarithms by Mr. Hansen

John Napier John Napier, a 16 th 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 16 th century arithmetic was barely up to the task and Napier became interested in this problem. Nicolaus Copernicus ( )

Even the most basic astronomical arithmetic calculations are ponderous. Johannes Kepler ( ) filled nearly 1000 large pages with dense arithmetic while discovering his laws of planetary motion! Johannes Kepler ( ) A typical page from one of Kepler’s notebooks

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: Common logs are based on ten raised to a power: x = 10 y Natural logs, which are based on the number e raised to a Natural logs, which are based on the number e raised to a power, are used mostly in higher and theoretical mathematics: x = e y x = e y 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. 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 : For logarithmic functions we use the notation : log a (x) or log a x This is read “log, base a, of x.” Thus, This is read “log, base a, of x.” Thus, y = log a x means x = a y And so a logarithm is simply an exponent of some base. 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. Logarithms are a mathematical tool originally invented to reduce arithmetic computations. Multiplication and division are reduced to simple addition and subtraction. Multiplication and division are reduced to simple addition and subtraction. Exponentiation and root operations are reduced more simple exponent multiplication or division. Exponentiation and root operations are reduced 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 logarithm functions for base 10 (common) and base e (natural) logs. Both log types can be used for ordinary calculations. 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: Exponential functions always have the variable in the exponent: f(x) = 2 x is an exponential function f(x) = x 2 is not an exponential function Definition: The function f(x) = a x, where a is a positive number constant other than 1, is called an exponential function, base a. Definition: The function f(x) = a x, 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 = log 3 x: 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 a 0 (a<>0) = 1, the graph of y = log a x, 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 Law for Multiplication b x ∙ b y = b x + y Law for Division Law for Division b x / b y = b x – y Law for Power of a Power Law for Power of a Power (b x ) y = b x y Law for Negative Exponents Law for Negative Exponents b -x = 1 / b x b -x = 1 / b x

Exponential and Logarithmic Relationships Conversion between log and exponential forms is often a convenient way to solve problems. Because x = a y and y = log a x are equivalent, then: 2 x = 8 is the same as x = log 2 8 Log problems are solved the same way: log 2 x = -3, the equivalent of 2 -3 And by the laws of exponents, we obtain: x = 1/2 3 or x = 1 / 8 x = 1/2 3 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 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 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. 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. 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, log a (xy) = log a x + log a y Let b = log a x and c = log a y. This is equivalent to: x = a b and y = a c Now multiply x and y: xy = a b a c, or, by a law of exponents, = a b + c As a logarithmic statement we can now write: log a (xy) = b + c log a (xy) = b + c And replacing the values of b and c, our solution is: log a (xy) = log a x + log a y

Additional Log Computations Other log theorems show these relationships: log a x p = p ∙ log a 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. log a x / y = log a x - log a y - divide two numbers by subtracting the logs of the two numbers. log b N = log a N / log a b - change the base of number N by dividing its log by the log of the new base. log b N = log a N / log a 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. Logarithm calculations produce answers as an exponent. To find the actual numeric solution of the calculation, the “antilogarithm” of the result must be found. 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. 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 Example: Using common logs, find the value of Step 1: Using a calculator, find the common logarithm of 37 ( ), then multiply that by 3, ( ). Step 2: Use a calculator to find the value of 10^ The answer is

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. 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: Determine the characteristic of these numbers: 37 = 3.7 ∙ 10 1 ; the characteristic is = 1.43 ∙ 10 2 ; 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 The log of 37 is thus Using the table, find the number 37 in the left-most column, and read its value in the second (0 column) as The log of 37 is thus You can check this with your calculator by computing which is You can check this with your calculator by computing which is

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 (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 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 (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 Now add the two logs together: = This is the logarithm of our answer. Now add the two logs together: = This is the logarithm of our answer. The answer, ( 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. The answer, ( 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. 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 The result, 5290, is one less than the actual answer of 5291, but is within 99.98% of the actual value. The final answer is calculated as 5.29 times 10 raised to the power of the characteristic 3, or 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. 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 ( ) 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. Henry Briggs ( ) 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. 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. 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: 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) = Now do the same, this time with the number 143. Now do the same, this time with the number 143. LOG(143) = Press the 2 nd button followed by the 10 x (LOG) button, then enter the sum of these two numbers ( ). Press the 2 nd button followed by the 10 x (LOG) button, then enter the sum of these two numbers ( ). The result, , 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 % of the actual value. The result, , 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 % of the actual value.

Conclusion Logarithms were originally devised to simplify arithmetic. 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 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. 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. 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.