Properties of Logarithmic Functions

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Logarithmic functions are the inverse of exponential functions where the function is 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: Given logx 16 = 4, find the value of the base ‘x’.

Solution: The given equation is logx 16 = 4

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

logb (a) = N; a = bN

Hence logx 16 = 4 can be written as 16 = x4

Now we prime factorization of 16 = 2 * 2 * 2 * 2

Therefore, 16 = 24. This gives 16 = x4; 24 = x4.

Hence x = 2.

Example 2: For the equation log3 (x2) = 2, then solve for ‘x’.

Solution: The given equation is log3 (x2) = 2.

According to the formula, we have log (am) = m * log a

Applying the above formula, we get

log3 (x2) = 2; 2 * log3 (x) = 2

Dividing by ‘2’ on both sides; log3 (x) = 2/2; log3 (x) = 1

Now using the formula, logb (a) = N; a = bN.

We get, x = 31; x = 3.

Hence x = 3 is the solution.