
Having trouble
1)An old coin was worth 50 cents originally and has been growing exponentially in value by 15% every year. Predict the coin's value after 3.5 years
2) Two investments are made. In one, $2500 is invested at 5% compounded annually. In other, $2000 is invested at 6% compounded annually. When will the investments have the same value?

#1:
Since the coin increases $0.15/yr, then after the first year it is worth $0.575.
$\displaystyle 0.575=0.50e^{k}$
$\displaystyle k=ln(1.15)$
$\displaystyle V=0.50e^{ln(1.15)t}$

Using the forumula $\displaystyle P=P_0(1+\frac{r}{n})^{nt}$
we get the two equations
$\displaystyle P=2500(1.05)^{t}$ and
$\displaystyle P=2000(1.06)^{t}$
using substitution we get the equation
$\displaystyle 2500(1.05)^{t}=2000(1.06)^t$
$\displaystyle (1.05)^t=\frac{4}{5}(1.06)^t$ taking ln of both sides
$\displaystyle ln(1.05)^t=ln(\frac{4}{5}(1.06)^t)$ using log properties
$\displaystyle tln(1.05)=ln(4/5)+tln(1.06)$ isolating t
$\displaystyle t(ln(1.05)ln(1.06))=ln(4/5) $
$\displaystyle t=\frac{ln(0.8)}{(ln(1.05)ln(1.06))}$