Figure out what to multiply a number by to get it to double after x times.

Question

I'm trying to figure out if there is a way to calculate what to multiply a number by to get it to double after $x$ times. For example, the number $100$, would be $200$ after multiplying it by $m$ $7$ times. I'm not sure of a way to do this aside from trying to manually find a number, but that is tedious and inaccurate. Any help would be appreciated, thanks.

Answer 1

Multiplying 7 times by $m$ means multiplying by $m\times m\times m\times m\times m\times m\times m = m^7$. If this doubles $x$, then $m^7\times x = 2x$ so, if $x\neq 0$, we have $$m^7 = 2,\;\;\text{i.e.}\;\; m = \sqrt[7]{2} = 2^{1/7} \simeq 1.104.$$ Note that this does not depend on the number $x$ you want to double.

Now, more generally, if you want to double a number by multiplying it $x$ times by $m$, then $$m^x = 2,\;\;\text{i.e.}\;\; m = \sqrt[x]{2} = 2^{1/x}.$$

Answer 2

Assume $x$ is the original number, which when multiplied by $m$, $y$ times, gets doubled.

$x×m^y=2x\\ m^y=2\\ y\log(m)=\log(2)\\ \log(m)=\dfrac{\log(2)}{y}\\ m=10^\left({\dfrac{log(2)}{y}}\right)$