Question

what is the inverse of the function $y= 4^{log _{ e^{x}$, (y>0)

Answer 1

$y=4^{\log_ex}\iff \log_e x=\log_4y$
$\iff x=e^{\log_4y}$

swap $x$ and $y$, the inverse function is
$y=e^{\log_4x}$

Answer 2

$y = 4^{log_e x},\ \ \ take\ logarithm\ to\ base\ 4\\\\log_4 y=log_e\ x,\ \ \ x=e^{log_4 y}$

It is possible to express x in terms of y, in another way.

$log_4 y = log_e x= log_4 x\ *\ log_e 4\\\\log_4 x=\frac{1}{log_e 4}\ log_4 y=log_4 e*log_4\ y=log_4\ y^{log_4 e}\\\\ So\ x = y^{log_4 e}=y^{0.7213..}$

y = f(x)  then  x = f⁻¹ (y)        , here f⁻¹ is called the inverse function.

Inverse function is obtained by interchanging x and y, after solving for x.

So inverse function is:
     $y = e^{log_4 x}\ \ \ or,\ \ \ y=x^{log_4 e}=x^{0.7213..}$
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