Logarithm is a shortcut

•Multiplication is a shortcut for addition. Recall that means 5 + 5+ 5. Exponents are a shortcut for multiplication. Recall that means . Logarithm is a shortcut for exponents.


Function

•Before we review exponential and logarithmic functions, let's review the definition of a function and the graph of a function. A function is just a rule. The rule links one number to a second number in an orderly and specific manner. All the points on the graph of a function are made up of two parts: (a number, and the function value at that number). For example, the amount of time needed to build a house could be the first number, the number of house built can be the function value. If a house needs a few months, then the rule is the amount of house multiply by the amount of time needed to build a house.

What is Logarithm

•A LOGARITHM is an exponent. It is the exponent needed to produce a certain number.
•Since
•23 = 8,
•then 3 is called the logarithm of 8 with base 2. We write
•3 = log28.
•We write the base 2 as a subscript.
•3 is the exponent to which 2 must be raised to produce 8.
•Since
•104 = 10,000
•then
•log1010,000 = 4.
•"The logarithm of 10,000 with base 10 is 4."
•4 is the exponent to which 10 must be raised to produce 10,000.
•"104 = 10,000" is called the exponential form.
•"log1010,000 = 4" is called the logarithmic form.


•Here is the definition:
•logbx = n means bn = x. That base with that exponent produces x.
•Example 1. Write in exponential form: log232 = 5
• 25 = 32
• Example 2. Write in logarithmic form: 4−2 = 116 . log4 1
•16 = −2. Example 3. Evaluate log81.
• 8 to what exponent produces 1? 80 = 1.
•log81 = 0.

•We can observe that in any base, the logarithm of 1 is 0.
•logb1 = 0
•Example 4. Evaluate log55.
• 5 with what exponent will produce 5? 51 = 5.
•log55 = 1.

In any base, the logarithm of the base itself is 1.
logbb = 1
Example 5. log22m = ?
2 raised to what exponent will produce 2m ? m, obviously.
log22m = m. The following is an important formal rule, valid for any
base b:

The following is an important formal rule, valid for any base b:
logbbx = x
This rule embodies the very meaning of a logarithm. x -- on the right -- is the exponent to which the base b must be raised to produce bx. The rule also shows that the inverseof the function logbx is the exponential function bx.


•Example 6 . Evaluate log3 19 . Answer. 19 is equal to 3 with what exponent? 19 = 3−2. log3 19 = log33−2 = −2. Compare the previous rule.
•Which numbers, then, will have negative logarithms?
•Proper fractions


•Example 7. log2 .25 = ?
25 = ¼ = 2−2. Therefore,log2 .25 = log22−2 = −2
•Example 8. log3 = ?
= 31/5. (Definition of a rational exponent.) Therefore,log3 = log331/5 = 1/5.


Natural logarithms

•The system of natural logarithms has the number called "e" as its base. (e is named after the 18th century Swiss mathematician, Leonhard Euler.) e is the base used in calculus. It is called the "natural" base because of certain technical considerations.
•ex has the simplest derivative

•e can be calculated from the following series involving factorials:
•e = 1 + 11! + 12! + 13! + 14! + . . . e is an irrational number, whose decimal value is approximately2.71828182845904.
•To indicate the natural logarithm of a number, we use the notation "ln."
•ln x means logex.


The Laws of logarithms

•1. logbxy = logbx + logby "The logarithm of a product is equal to the sumof the logarithms of each factor."
•2. logb xy = logbx − logby "The logarithm of a quotient is equal to the logarithm of the numeratorminus the logarithm of the denominator."
•3. logb x n = n logbx "The logarithm of a power of x is equal to the exponent of that power times the logarithm of x."


Examples

•Use the laws of logarithms to rewrite log z5 . Answer. According to the first two laws,
•log z5 = log x + log − log z5 Now, = y½. Therefore, according to the third law,
•log z5 = log x + ½ log y − 5 log z

•Use the laws of logarithms to rewrite ln .
• Solution.
•ln = ln (sin x ln x)½ = ½ ln (sin x ln x), 3rd Law = ½ (ln sin x + ln ln x), 1st Law Note that the factors sin x ln x are the arguments of the logarithm function


•Solve this equation for x:
•log 32x + 5 = 1 Solution. According to the 3rd Law, we may write (2x + 5)log 3 = 1 Now, log 3 is simply a number. Therefore, on multiplying out by log 3, 2x· log 3 + 5 log 3 = 1 2x· log 3 = 1 − 5 log 3 x = 1 − 5 log 3 2 log 3 By this technique, we can solve equations in which the unknown appears in the exponent.


Proof of the laws of logarithms


•The laws of logarithms will be valid for any base. We will prove them for base e, that is, for y = ln x.
•1. ln ab = ln a + ln b.
•The function y = ln x is defined for all positive real numbers x. Therefore there are real numbers p and q such that
•p = ln a and q = ln b.
•This implies
•a = e p and b = e q.
•Therefore, according to the rules of exponents,
•ab = e p· e q = ep + q.
•And therefore
•ln ab = ln ep + q = p + q = ln a + ln b.
•That is what we wanted to prove.

•In a similar manner, 2nd law can be prove.
•3. ln an = n ln a.
•There is a real number p such that
•p = ln a;
•that is,
•a = e p.
•And the rules of exponents are valid for all rational numbers n. (Lesson 29 of Algebra; an irrational number is the limit of a sequence of rational numbers.) Therefore,
•an = e pn.
•This implies
•ln an = ln e pn = pn = np = n ln a.
•That is what we wanted to prove.

History of Logarithms

It is not known exactly when logarithms were first invented, with evidence of use in 8th century India. However, their invention as an aid to calculations is attributed to John Napier in the early 17th century. He released the first ever log tables in 1614.
JOOST BURGI
Another mathematician, Joost Burgi from Prague, had been working independantly on logarithms at the same time as Napier but had a slightly different approach. He released his log tables in 1920. These took the form of a comparison between a geometric series and an arithmetic series, which clearly show the operation and nature of logs. A very simple example of this can be seen in the table below:








This table shows the geometric series on top and the arithmetic along the bottom. To multiply two numbers on the top, add the two numbers directly below and find the corresponding number on top. For example, to multiply 1/2 and 8, the numbers below (in the arithmetic sequence) are -1 and 3 respectively. Adding these is equal to 2 and the corresponding number (in the geometric sequence) for 2 is 4 (the correct answer.) This process also works for division (subtract the two numbers), and raising 2 to a power (find the power in the arithmetic sequence and look above.) This works because the numbers on top are the number 2 to the power of the corresponding numbers on the bottom. These logarithms are therefore to the base 2. This is how arithmetic processes were made easier by logarithms.
John Napier and Henry Briggs
John Napier was a famous Scottish theologian and mathematician who lived from 1550 to 1617. He was a very intelligent man, considered by many of the locals to be in league with the devil. As well as being educated, Napier was also a baron, nobelman, the 7th Laird of Merchiston and owner of a considerable estate. He defined his logarithms in terms of relative rates, in a different way to Joost.




















After the release of Napier's logarithm tables, Henry Briggs, an English mathematican who at the time had been researching astronomy and its applications to navigation, immediately saw the potenial for logarithms to ease astronomical and navigational calculations and so turned his attention to developing the idea. During 1915 and 1916 Briggs travelled to Edinburgh to collaberate with Napier whereby they modified Napier's logarithms, which at the time were to the base 1/e, to the base 10. These were easier to work with and would be more useful for Briggs' astronomical and navigational calculations. Logarithms to the base 10 are known as common logarithms or Briggsian logarithms in his honour. In 1924 Briggs published the "Arithmetica Logarithmica" (common logarithms) which included tables of logs from 1 to 20,000 and 90,000 to 100,000 to 14 decimal places, as well as information about the nature and constuction of logs