Class 12 Maths Continuity Differentiability Exponential and Logarithmic Functions

Exponential and Logarithmic Functions

The exponential function with positive base b > 1 is the function

y = f(x) = bx

Let b > 1 be a real number. Then we say logarithm of a to base b is x if

bx = a

Logarithm of a to base b is denoted by logb a.

Hence, logb a = x if bx = a.

Properties of ‘log’ functions:

  1. loga p = logb p/ logb a
  2. logb pq = logb p + logb q
  3. logb pn = n logb p
  4. logb p/q = logb p/logb q

Problem: Differentiate the following w.r.t. x: ex/sin x

Solution:

Let y = ex/sin x

Differentiate w.r.t. x, we get

dy/dx = d(ex/sin x)/dx

By using the quotient rule, we obtain

     dy/dx = {sin x * d(ex)/dx -  ex * d(sin x)/dx}/(sin x)2

=> dy/dx = {sin x * ex - ex * cos x}/sin2 x

=> dy/dx = ex(sin x - cos x)/sin2 x, x ≠ nπ, n є Z

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