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# (a) How long does it take for an investment to double in value if it is invested at 8% compounded monthly?(b) How long does it take if the interest is compounded continuously?

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(a) If we want to double an investment, the future value $A$ will be equal to $2P$, where $P$ is the present value. To calculate the time we need to double an investment we will apply formula for future value, which is

$A=P\cdot\left(1+\frac{r}{n}\right)^{nt},$

where $t$ is time expressed in years, $r$ is per annum interest rate compounded $n$ times per year.We know that $A=2P, r=8\%=0.08$ and $n=12$ because we compound interest monthly, so substituting this into the previous equation we will get

$2P=P\left(1+\frac{0.08}{12}\right)^{12\cdot t}.$

Dividing the previous equation by $P$ we will get

\begin{align*} 2=\left(\frac{1.08}{12}\right)^{12t}\approx &1.0067^{12t}\\ =&(1.0067^{12})^t\\ \approx &1.083^t \end{align*}

Using the definition of the logarithm we have now that

\begin{align*} 2=(1.083)^t\quad \Longleftrightarrow\quad &t=\log_{1.083}2\\ \Longrightarrow\quad & t=\frac{\ln 2}{\ln 1.083}\\ \Longrightarrow\quad & t\approx 8.69314. \end{align*}

We used the Change-of-Base formula for logarithms in the last step. So, the time needed to double the investment if it is invested at $8\%$ compounded monthly is approximately 8.69 years.

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