LOGARITHM -- NEED OF EXTENSION OF EXPONENTS, DEFINITION, FUNCTION, LINKS TO FURTHER STUDY

Before you proceed , Look at the need of Logarithm in Exponents,
explained at the end, by means of an example,if you have not already done so. Also knowledge of Exponents is a prerequisite here.

Look at a question in exponents: what is x if 7^{x} = 343. The same question can be asked here by using a symbol.The question is " what is log_{7} 343 ? " If you understand this algebraic symbol, which you will, a little later, you will understand the question as find x such that 7^{x} = 343. More over while solving the question, you will write 7^{x} = 343 = 7^{3} ⇒ x = 3. Since 343 is 7^{3}, you could easily answer it. If I give you find x such that 7^{x} = 100, what will you do? With the symbolism you learn here, you will answer it as x = log_{7} 100. It is not only the symbolism for finding the exponents, but a useful body of knowledge, you will learn here.

Definition of Logarithm :

If a(≠1) and n are any positive real numbers
and for some real x, a^{x} = n,
then x is said to be the logarithm of n to the base a.
It is written as log_{a}n = x.

i.e.

For real values of a, x, n and a≠1, a > 0, n > 0 a^{x} = n ⇒ log_{a}n = x

Examples:
3^{4} = 81 ⇒ log_{3} 81 = 4
9^{2} = 81 ⇒ log_{9} 81 = 2
From these examples we can conclude that the logs of the same
number (here 81) to different bases (here 3 and 9) are different (4 and 2). i.e.

The logs of the same number to different bases are different.

Some more examples:
64^{1⁄3} = 4 ⇒ log_{64} 4 = 1⁄3
5^{-2} = 1⁄25 ⇒ log_{5} (1⁄25) = -2

You may note the following points.

Base of the log (a in definition) is positive and not equal to 1.

Here logs are defined only for positive real numbers (n in definition).

The value of the log (x in definition) can be negative.

a^{x} = n is called exponential form.

log_{a}n = x is called logarithmic form.

Both in exponential form and logarithmic form, the base is same.

The exponent in exponential form is the value of the log
(to the same base to which the exponent is raised) in logarithmic form.

Also there exists a unique x which satisfies the equation
a^{x} = n. So log_{a}n is unique.

Get The Best Grades With the Least Amount of Effort : Logarithm

Here is a collection of proven tips, tools and techniques to turn you into a super-achiever - even if you've never thought of yourself as a "gifted" student.

The secrets will help you absorb, digest and remember large chunks of information quickly and easily so you get the best grades with the least amount of effort.

If you apply what you read from the above collection, you can achieve best grades without giving up your fun, such as TV, surfing the net, playing video games or going out with friends!

Look at the following two sets of examples 1st set: 2^{3} = 8 ⇒ log_{2} 8 = 3 2^{4} = 16 ⇒ log_{2} 16 = 4 2^{5} = 32 ⇒ log_{2} 32 = 5 Here as the number (8, 16, 32) increases,the log of the number (3, 4, 5) increases. Note that the base (2) is more than 1. What is seen to be true here in this example, is infact true in general. The general statement is

If a > 1, n_{1} > n_{2} ⇒ log_{a}n_{1} > log_{a}n_{2} i.e. when the base is more than 1, the logarithmic function is an increasing function.

2nd set: (1⁄2)^{5} = 1⁄32 ⇒ log_{(1⁄2)} (1⁄32) = 5 (1⁄2)^{4} = 1⁄16 ⇒ log_{(1⁄2)} (1⁄16) = 4 (1⁄2)^{3} = 1⁄8 ⇒ log_{(1⁄2)} (1⁄8) = 3 Here as the number (1⁄32, 1⁄16, 1⁄8) increases,the log of the number (5, 4, 3) decreases. Note that the base (1⁄2) is less than 1. What is seen to be true here in this example, is infact true in general. The general statement is

If 0 < a < 1, n_{1} > n_{2} ⇒ log_{a}n_{1} < log_{a}n_{2} i.e. when the base is less than 1 (and positive), the logarithmic function is a decreasing function.

Research-based personalized Math Help tutoring program : Logarithm

Here is a resource for Solid Foundation in Math Fundamentals from Middle thru High School. You can check your self by the

Are you spending lot of money for math tutors to your child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and more personalized Math Help to identify and fix the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend and that too With a minuscule fraction of the amount you spent on tutors with unconditional 100% money back Guarantee, if you are not satisfied.

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program that identifies exactly where your child needs help and then creates a personal instruction plan tailored to your child’s specific needs.

If your child can use a computer and access the Internet, he or she can use the program. And your child can access the program anytime from any computer with Internet access.

There is an exclusive, Parent Information Page provides YOU with detailed reports of your child’s progress so you can monitor your child’s success and give them encouragement. These Reports include

Time spent using the program

Assessment results

Personalized remediation curriculum designed for your child

Details the areas of weakness where your child needs additional help

Provides the REASONS WHY your child missed a concept

List of modules accessed and amount of time spent in each module

Quiz results

Creates reports that can be printed and used to discuss issues with your child’s teachers

These reports are created and stored in a secure section of the program, available exclusively to you, the parent. The section is accessed by a password that YOU create and use. No unauthorized users can access this information.

Its research-based results have proven that it really works for all students! in improving math skills and a TWO LETTER GRADE INCREASE in math test scores!,if they invest time in using the program.

Proven for More than 10,000 U.S. public school students who increased their math scores.

Recently, I have found a series of math curricula (Both Hard Copy and Digital Copy) developed by a Lady Teacher who taught everyone from Pre-K students to doctoral students and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous over many of the traditional books available. These give students tools that other books do not. Other books just give practice. These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts from their existing knowledge. These provide many pages of practice that gradually increases in difficulty and provide constant review.

These also provide teachers and parents with lessons on how to work with the child on the concepts.

The series is low to reasonably priced and include