Exam of Logarithms in General (Nandan Nayak)

Introduction to exam of logarithms in general:

Exam of logarithm in general involves the learning and understanding of the following concepts of logarithms:-

Conversion of exponential to logarithmic form

The three laws of logarithms

The base changing formula

After learning the above topics of logarithms, the exam of logarithms in general preparation can be considered almost complete at the basic level. Essentially, one gains the application knowledge of logarithmic laws and concepts that enable one to simplify and solve logarithmic equations. The above basic logarithmic concepts help us to prove advanced logarithmic statements.

The above mentioned concepts on logarithms are briefly explained below as preparation for exam on logarithms in general.

Conversion of exponential to logarithmic forms in general:

The logarithm of a number is equal to the exponent to which the base of that logarithm must be raised in order to obtain that number.

Logarithms can be considered as a different type of representation of exponential statements.

For example, the exponential statement '6^2 = 36' can be expressed in the logarithmic form as follows:-

log '(6) 36 = 2'

In the above conversion from exponential to logarithmic form, we note the following rules:-

Base of the logarithm and the exponential form is the same.

The result in the exponential form is made the object in the logarithmic form.

The exponent in the exponential form is made the result in the logarithmic form.

The three laws of logarithms:

The three standard laws of logarithms can be stated as follows:-

First law of logarithms - the product law

The logarithm of a product of two or more numbers is equal to the sum of the logarithms of each of the numbers in the product. Thus, if 'a' and 'b' are two non-negative real numbers, and 'c' is the base of the logarithms, then,

'log (c) ab = log (c) a + log (c) b'

This law highlights that logarithms reduce multiplication to addition.
Second law of logarithms - the quotient law

The logarithm of the quotient of two numbers is equal to the difference of the logarithms of each of the two numbers. Thus, if 'a 'and 'b' are two non-negative real numbers, and 'c' is the base of the logarithms, then,

'log (c) a/b = log (c) a - log (c) b'

This law highlights that logarithms reduce division to subtraction.
Third law of logarithms - the power law

This law defines the logarithm of an exponential expression. The logarithm of an exponential expression is equal to the product of the exponent in the exponential expression and the logarithm of the base of that exponential expression.

'Log (c) a^b = b * log (c) a'

This law highlights that logarithms reduce exponents to products.

The base changing formula

The base changing formula helps us to change the base of logarithms, which is a very essential function in solving logarithmic equations. It helps to simplify logarithmic equations. By using the base changing formula, one can change the base of a logarithm to any other number or variable.

'Log (a) b = (log (c) b)/(log (c) a)'

In the above statement, the base of the logarithm is being changed from 'a' to 'c'.

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