Which of the following formula represents a first order reaction

Consider the reaction $\text{A}\to \text{Products}$

Let $a$  is the concentration of A at initial.

After time t, the concentration of A

Becomes $(a-x)$ and $x$ has been changed into products.

 The rate reaction after time ‘t’ is given by

          $\frac{dx}{dt}=k(a-x)$

        $\Rightarrow \frac{dx}{(a-x)}=kdt$

Upon integration of above equation, will get

     ${\displaystyle \int \frac{dx}{(a-x)}=k{\displaystyle \int dt}}$

$-{\log }_{\text{e}}(a-x)=kt+\text{c}\to \text{eqn(1)}$

Where c is integration constant

When        $t=0x=0$

  $\text{C}=-{\log }_{e}a$

Putting the value of c in equation (1), we will get

${\log }_{e}a-{\log }_e(a-x)=kt$

$\Rightarrow {\log }_e\frac{a}{(a-x)}=kt$

$k=\frac{2.303}{t}{\log }_{10}\frac{a}{(a-x)}=\frac{2.303}{t}{\log }_{10}\frac{{[\text{A]}}_0}{{\left[\text{A}\right]}_t}$

Here , k is rate constant , ${\left[\text{A}\right]}_{\text{0}}=a\text{and}{\left[\text{A}\right]}_{\text{t}}=a-x$