Question

The value of an investment is expected to grow by 5% per year.

How long will it take for the value to double?

Answer 1

hmmm let's start with 1 buck, once we double that, it'd become 2 bucks, so how long for $1 to be $2?

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill &\$2\\ P=\textit{initial amount}\dotfill &\$1\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ t=years\dotfill &t\\ \end{cases} \\\\\\ 2=1(1 + 0.05)^(t)\implies 2=1.05^t\implies \log(2)=\log(1.05^t) \\\\\\ \log(2)=t\log(1.05)\implies \cfrac{\log(2)}{\log(1.05)}=t\stackrel{\textit{about 14 years, 2 months and a half}}{\implies 14.21\approx t~\hfill }`