# Properties of Logarithms

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Logarithmic are the inverse of exponential functions. They are commonly used in many branches in math. Logarithms are written as f(x) = log b (a) such that ‘b’ > 0, b = 1 and a > 0. This is read as “log base b of a”. Logarithmic functions have many properties and rule which are used to solve many questions:

General properties (where x> 0, y>0)

logb (xy) = logbx + logb y

logb (x/y) = logbx  - logby

logb (xm) = m logb x

logb b = 1

Example 1: Solve logx 27 = 3, find the value of the base ‘x’.

Solution: The given equation is logx 27 = 3

Convert this Logarithmic equation to Exponential equation by using the formula,

logb (a) = N; a = bN

Hence logx 16 = 4 can be written as 27 = x3

Now we prime factorization of 27 = 3 * 3 * 3.

Therefore, 27 = 33. This gives 27 = x3; 33 = x3.

Hence x = 3 is the solution.

Example 2: Solve logx 225 = 2, find the value of the base ‘x’.

Solution: The given equation is logx 225 = 2.

Convert this Logarithmic equation to Exponential equation by using the formula,

logb (a) = N; a = bN

Hence logx 225 = 2 can be written as 225 = x2.

Now we prime factorization of 225 = 3 * 3 * 5 * 5.

Therefore, 225 = 152. This gives 225 = x2; 152 = x2.

Hence x = 15 is the solution.