# PROPERTIES OF LOGARITHMS -- PROOFS AND EXPLANATIONS OF FORMULAS, LINKS FOR FURTHER STUDY

Introduction to Logarithms before Properties of Logarithms

if you have not already done so.

There we discussed the need for extension
of Exponents and introduction to the new
branch of study called Logarithms.

It is a prerequisite here.

## Proofs and Explanations for Formulas in Logarithms :

### Proof and Explanation of Formula 1 : Properties of Logarithms

Formula from definition of Logarithm:

ax = n ⇔ loga n = x

This formula is directly from
definition of Logarithm.

This formula from
exponential form to logarithmic form
and from
logarithmic form to exponential form
is useful in solving many a problem.

The student should be
thoroughly familiar with it
in applying in both directions.

In remembering this formula,
the following two points are helpful.
• Both in exponential form and
logarithmic form, the base is same.
• The exponent in exponential form
is the value of the logarithm
in logarithmic form.

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.

and remember large chunks of information
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!

Know more about the Speed Study System.

### Proof and Explanation of Formula 2 : Properties of Logarithms

Logarithm as an Exponent to its Base:

a(loga n) = n

Proof:
From the definition of logarithm, we have
if ax = n..............(i)
then x = loga n ................(ii)
Substituting the value of x from (ii) in (i), we get
a(loga n) = n (Proved.)

In remembering this formula,
you have to observe that the
base of the exponential form
is same as the base of the
logarithm in the exponent.

### Proof and Explanation of Formula 3 : Properties of Logarithms

Logarithm of 1 to any Base:

loga 1 = 0

Proof:
From Laws of Exponents, we know a0 = 1
⇒ loga 1 = 0 (Proved.)

It is easy to remember Logarithm of 1 to any Base is zero.

Great Deals on School & Homeschool Curriculum Books

## Research-based personalized Math Help tutoring program : Properties of Logarithms

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

### FREE TRIAL.

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.

### SUBSCRIBE, TEST, IF NOT SATISFIED, RETURN FOR FULL REFUND

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.

### Unique program to help improve math skills quickly and painlessly.

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.

### Personalized remediation curriculum designed for your child

Thus The features of this excellent Tutoring program are

• Using detailed testing techniques
• Identifing exactly where a student needs help.
• Its unique, smart system pinpointing precise problem areas -
• slowly and methodically guiding the student
• raising to the necessary levels to fix the problem.

### Not a “one-size-fits-all” approach!

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.

### Proof and Explanation of Formula 4 : Properties of Logarithms

Logarithm of any number to the same Base:

loga a = 1

Proof:
From Laws of Exponents, we know a1 = a
⇒ loga a = 1 (Proved.)

It is easy to remember Logarithm of any number to the same Base is one.

### Proof and Explanation of Formula 5 : Properties of Logarithms

Logarithm of a Product:

loga (mn) = loga m + loga n

Proof:
Let loga m = P ⇒ aP = m ............(i)
Let loga n = Q ⇒ aQ = n .............(ii)
You might have observed that
Equations (i) and (ii)
are obtained by changing
the Logarithmic form to Exponential form.

(i) x (ii) gives aP x aQ = mn
aP + Q = mn ( Since aP x aQ = aP + Q From laws of Exponents)
⇒ loga (mn) = P + Q ( by changing Exponential form to Logarithmic form)
By Replacing the values of P and Q, we get
loga (mn) = loga m + loga n (Proved.)

In proving this, see how we made use of
changing Logarithmic to Exponential form
and Exponential to Logarithmic form.

Remember that Logarithm of a Product
is the sum of the Logarithms of
the Factors of the Product.

Remember that the Formula is not for log(m + n) nor for log m x log n.

We have Formula for log (mn) and log m + log n.

We should be able to apply the formula
from L.H.S. to R.H.S. and from R.H.S. to L.H.S.

### Proofs and Explanations of Formula 6 to 10 : Properties of Logarithms

The following Link takes you to the
Proofs and Explanations of Formula 6 to 10.

Proofs and Explanations of Formula 6 to 10

## Progressive Learning of Math : Properties of Logarithms

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

Elementary Math curriculum

and

Algebra Curriculum.