I am currently working my way through Hughes-Hallet et al., Calculus- Single and Multivariable.
I am having trouble with the following problem.
Show algebraically that if $P=P_0a^t$ doubles between time $t$ and time $t + d$, then $d$ is the same number for any $t$.
I have started by writing out two equations.
$$ P=P_0 a^{t}$$
$$ 2P=P_0a^{(t+d)}$$
Not sure where to go from here.
$P=P_o a^{t}$ .... (1)
$2P=P_oa^{(t+d)}$ .... (2)
If we divide (2) by (1) and simplify, we will get $d = {log(2) \over log(a)}$ , which is constant if a is given as constant.