b p = log p An easy generalisation of this (left as an exercise!) is log b p n = n log p for any positive integer n . In fact this is true for any real number n , but we will not attempt to prove this. On the similar lines the reader is invited to verify log b y = log b x – log b y Example Is it true that x = e log x for all real x ? Solution First, observe that the domain of log function is set of all positive real numbers.
So the above equation is not true for non-positive real numbers. Now, let y = e log x . If y > , we may take logarithm which gives us log y = log ( e log x ) = log x . log e = log x .
Thus y = x . Hence x = e log x is true only for positive values of x . One of the striking properties of the natural exponential function in differential calculus is that it doesn’t change during the process of differentiation. This is captured in the following theorem whose proof we skip.
Theorem * ( ) The derivative of e x w.r.t., x is e x ; i.e., d dx ( e x ) = e x . ( ) The derivative of log x w.r.t., x is x ; i.e., d dx (log x ) = x . Example Differentiate the following w.r.t. x : (i) e – x (ii) sin (log x ), x > (iii) cos – ( e x ) (iv) e cos x Solution (i) Let y = e – x .
Using chain rule, we have − ⋅ (– x ) = – e – x (ii) Let y = sin (log x ). Using chain rule, we have cos (log ) cos (log ) (log ) * Please see supplementary material on Page . (iii) Let y = cos – ( e x ). Using chain rule,