LOGARITHMS -- DEFINITION, FORMULAS WITH PROOFS, SOLVED EXAMPLES, EXERCISES

Please study

Introduction to 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.

It is a prerequisite here.

Here, we will study Various Formulae
and provide links to further study.

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Log Formulas :

In the following formulas, the base of the log is positive real number (≠ 1). When we write log of a literal, that literal is a positive real number.

Log Formula 1 :
Formula from definition of Log:

a^x = n ⇔ log_a n = x

Log Formula 2 :
Log as an Exponent to its Base:

a^{log_a n} = n

Log Formula 3 : Log of 1 to any Base:

log_a 1 = 0

Log Formula 4 :
Log of any number to the same Base:

log_a a = 1

Log Formula 5 :
Log of a Product:

log_a (mn) = log_a m + log_a n

Log Formula 6 :
Log of a Quotient:

log_a (m⁄n) = log_a m - log_a n

Log Formula 7 :
Log of a Power:

log_a (mⁿ) = n log_a m

Log Formula 8 :
Change of Base of a Log :

log_a b = (log_k b)⁄(log_k a)

Log Formula 9 :
Log of a Power to the Base of another Power:

log_aⁿ (b^m) = (m⁄n) log_a b

Log Formula 10 :
Log of b to the base a; b and a interchanged:

log_a b = 1⁄(log_b a)

All the above 10 Formulas are to be remembered and to be appliedto solve various problems. What we said about Formulas andproblems in Exponents holds good here also. We repeat that here.

The best way to remember various Algebra Formulas(or Math Formulas) is to apply them to a number of problems.The best way to solve Algebra Problems (or Math Problems) is to remember various formulas.

Proofs and Explanations of the Formulas :

The Proofs and Explanations for
the above Formulas in Logarithms
are provided at the following Links.

Proofs and Explanations for first 5 Formulas

Proofs and Explanations for 6 to 10 Formulas

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Solved Examples

The following Links take you
to the sets of Solved Examples.

Set 1 of Solved Examples

Set 2 of Solved Examples

Set 3 of Solved Examples

Set 4 of Solved Examples

Set 5 of Solved Examples

Logarithm Tables : Characteristic and Mantissa

To find the log value of a number using Tables
we need the knowledge of Charateristic and Mantissa
which is covered in Logarithm Tables.

Exercise : Logarithms

The above Five Sets of Examples
also contain Exercise problems.

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