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.