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# Properties of Logarithms

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Logarithm is the power of a number which increases the number in order to obtain the new number. The base 10 Logarithm of the value 1000 is 3, because the base 10 raised to the power of 3 is 1000. Because if we multiply the base value 3 times we get 1000.

10 * 10 * 10 = 1000

log 103 = 3 log 10 = 3

Logarithm is a special kind of function, whose functionality is defined in their properties. So, now we discuss properties of logarithms to understand the Logarithm function. The following are some of the Logarithm properties:

Property 1: Multiplication of two variables in log function can be represent as an addition of two log Functions.

log­b (x * y) = logb x + logb y

E.g, log­b (5 * 7) = logb 5 + logb 7

Property 2: Division of two variables in log function can be represent as a subtraction of two log Functions.
log­b ($\frac{x}{y}$) = logb x - logb y

E.g, log­b ($\frac{9}{6}$) = logb 9 - logb 6

Property 3: Power of a variables in log function can be represent as a multiplication between power and variable log.

log­b xn = n * logb x

E.g, log­b 53 = 3 * logb 5

Property 4: Base of variables in log function can be represent as,

log­b x = $\frac{\log_{a}{x}}{\log_{a} {b}}$

E.g, log­2 7 = $\frac{\log_{a}{7}}{\log_{a} {2}}$

There are following other properties in logarithm functions:
1. logb 1 = 0
2. logb b = 1
3. logb b2 = 2
4. logb bx = x
5. BlogB x = x
6. loga b = $\frac{1}{\log_{b} {a}}$