Question

differentiate a*x in terms of log​

Answer 1

We have to differentiate a^x with respect to x and express in terms of log.

$let \: y \: = {a}^{x}$

Now taking log both sides :-

$\implies log ( y ) = log( {a}^{x} )$

We know that :-

$\underline{\boxed{\bf{ log( {a}^{b} ) = b \times log(a) }}}$

Therefore :-

$\implies log ( y ) =x \times log( {a})$

Now differentiating both sides :-

$\implies \dfrac{d(log ( y ))}{dx} = \dfrac{d(x \times log( {a}))}{dx}$

we know that :-

$\underline{\boxed{\bf{ \frac{d( log(x)) }{dx} = \dfrac{1}{x} }}}$

$\implies \dfrac{1}{y} \dfrac{d y }{dx} = log( {a}) \times \dfrac{d(x )}{dx}$

log(a) is constant.

$\implies \dfrac{1}{y} \dfrac{d y }{dx} = log( {a}) \times 1$

$\implies \dfrac{1}{y} \dfrac{d y }{dx} = log( {a})$

$\implies \dfrac{d y }{dx} = y \: log( {a})$

now put y = a^x

$\implies \bf \: \dfrac{d y }{dx} = \bf {a}^{x} log( {a})$