Simple Chemical Conversion

From the results of chemical experimentation of substance converted into another substance, it was found that the rate of change of unconverted substance is proportional to the amount of unconverted substance.
 

If x is the amount of unconverted substance, then

with a condition that x = x_o when t = 0.
 

$\dfrac{dx}{dt} = -kx$

$\dfrac{dx}{x} = -k \, dt$

$\ln x = -kt + \ln C$

$\ln x = \ln e^{-kt} + \ln C$

$\ln x = \ln Ce^{-kt}$

$x = Ce^{-kt}$
 

From the initial condition of x = x_o when t = 0
$C = x_o$
 

Thus,