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

Please 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:

a^x = n ⇔ log_a n = x

• 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.

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Properties of Logarithms

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Proof and Explanation of Formula 2 :
Properties of Logarithms

Logarithm as an Exponent to its Base:

a^{(log_a n)} = n

Proof and Explanation of Formula 3 :
Properties of Logarithms

Logarithm of 1 to any Base:

log_a 1 = 0

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Properties of Logarithms

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Proof and Explanation of Formula 4 :
Properties of Logarithms

Logarithm of any number to the same Base:

log_a a = 1

Proof and Explanation of Formula 5 :
Properties of Logarithms

Logarithm of a Product:

log_a (mn) = log_a m + log_a n

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

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