What are the Properties of Logarithms

Here you will learn what are the properties of logarithms and fundamental identities of logarithm with examples.

Let’s begin –

Every positive real number N can be expressed in exponential form as \(a^x\) = N where ‘a’ is also a positive real number different than unity and is called the base and ‘x’ is called an exponent.

We can write the relation \(a^x\) = N in logarithmic form as \(log_aN\) = x. Hence \(a^x\) = N <=> \(log_aN\) = x. Hence logarithm of a number to some base is the exponent by which the base must be raised in order to get that number.

\(log_aN\) is defined only when

(i) N > 0

(ii) a > 0

(iii) \(a\neq1\)

Properties of Logarithms

If m, n are arbitrary positive numbers where a>0,\(a\neq1\) and x is any real number, then-

(a)  \(log_a mn\) = \(log_a m\) + \(log_a n\)

(b)  \(log_a\)\(m\over n\) = \(log_a m\) – \(log_a n\)

(c)  \(log_a\)\(m^x\) = x\(log_a m\)

Example : If \(a^2\) + \(b^2\) = 23ab, then show that \(log (a + b)\over 5\)= \(1\over 2\)(log a + log b).

Solution : \(a^2\) + \(b^2\) = \((a+b)^2\) – 2ab = 23ab
=> \((a+b)^2\) = 25ab
=> a+b = 5\(\sqrt{ab}\)

L.H.S. = \(log(a+b)\over 5\) = \(log(5 \sqrt{ab}) \over 5\) = \(1 \over 2\)log ab = \(1 \over 2\)(log a + log b) = R.H.S.

Fundamental Identities

Using the basic definition of logarithm we have 2 important deductions:

(a)  \(log_NN\) = 1   i.e  logarithm of a number to the same base is 1.

(b)  \(log_N\)\(1\over N\) = -1   i.e  logarithm of a number to the base as its reciprocal is -1.

Note :

N = \((a)^{\log_a N}\)     e.g. \(2^{\log_2 7}\) = 7

Example : If \(log_4m\) = 3,then find the value of m.

Solution : \(log_4m=3\) => \(m=4^3\) => \(m=64\).

Hope you learnt what are the properties of logarithms and fundamental identities of logarithm. To learn more practice more questions and get ahead in competition. Good Luck!

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