Lily pads

We know that if there are $n$ lily pads on the $k^{th}$ day,then there will be $2n$ lily pads on the $(k+1)^{st}$ day.

The crux move is using this relation "in reverse".If lily pads fill the pond in 50 days,then they will fill half the pond in 49 days.

there is one lily pad and the next day number of lily pads doubles so that mean $1 \times 2$ $=$ $2$ .But It takes 50 days to fill the pond so that mean $2^{50}$

because $2^{50}$ $=$ $2 \times 2 \times 2 \times 2 \times 2...$ $=$ $2^{50}=1125899906842624$ and than we need to know how many days does it take to fill half of the pond?

so we need to calculate the half $\frac{1125899906842624}{2}$ $=$ $562949953421312$ and we need to find an exponents that equal to that number $2^{x}=562949953421312$ Take log of both sides $log(2x)=log(562949953421312)$ $\rightarrow$ $x \times (log(2))=log(562949953421312)$ $\rightarrow$ $\frac{log(562949953421312)}{log(2)}$ $\rightarrow$ $x=\boxed{49}$