# 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!