PROPERTIES OF LOGARITHMS

Property 1 :

Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base.

log_amn = log_am + log_an

Property 2 :

Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base.

log_a(m/n) = log_am - log_an

Property 3 :

Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.

log_amⁿ = nlog_am

Property 4 :

If log_am  =  x, then

m = a^x

Property 5 :

If k  =  x^y, then

log_xk = y

(Properties 4 and 5 explain the relationship between indices and logarithms) 

Property 6 :

Multiplication of Logarithms :

Multiplication of two or more logarithms can be simplified, if one of the following two conditions is met.

Condition (i) :

The argument of the first logarithm and the base of the second logarithm must be same.

Condition (ii) :

The base of the first logarithm and the argument of the second logarithm must be same.

Example 1 :

Simplify : log_ab ⋅ log_bc

In the above two logarithms, the argument of the first logarithm and the base of the second logarithm are same. 

So, we can simplify the multiplication of above two logarithms by removing the part circled in red color. 

Example 2 :

Simplify : log_xy ⋅ log_zx

In the above two logarithms, the base of the first logarithm and the argument of the second logarithm are same.

So, we can simplify the multiplication of above two logarithms by removing the part marked in red color.

Property 7 :

Change of Base :

Multiplication of two or more logarithms can be simplified, if one of the following two conditions is met.

If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation.

log_ba = log_xa ⋅ log_bx

log_ba = log_xa / log_xb

More clearly, 

In the above example, the base of the given logarithm 'b' is changed to 'x'.

In this way, we can change the base of the given logarithm to any other base.

Property 8 :

Inverse Properties of Logarithms :

Property 9 :

If you switch a logarithm from numerator to denominator or denominator to numerator, we have to interchange the argument and base.

Examples :

log_ab = 1/log_ba

5/(2log_xy) = 5log_yx/2

Property 10 :

Logarithm of any number to the same base is equal to 1.

Examples :

log_xx = 1

log₃3 = 1

The above property will not work, if both the base and argument are 1.

log₁1 ≠ 1

The reason for the above result is explained in the next property.

Property 11 :

Logarithm of 1 to any base is equal to 0.

Examples :

log_x1 = 0

log₃1 = 0

log₁1 = 0

1.  If the base not given in a logarithm, it can be taken as 10.

log10 = log₁₀10 = 1

log1 = log₁₀1 = 0

Note :

Logarithm using base 10 is called Common logarithm and logarithm using base 'e' is called Natural logarithm.

Here, e  ≈  2.718 called exponential number.

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