A pair of shoes now costs $85 and the price is expected to increase 3% each year.Approximately how many years from now will the price be $100?
Note: A 3% increase in the price means you mulitply by 103%, or by 1.03.
This is essentially a geometric series. Let t be the number of years and P the price of the shoes in dollars. Then:
$\displaystyle P(t) = 85 \cdot (1.03)^t$
So, we want to know t when P = $100.
$\displaystyle 100 = 85 \cdot (1.03)^t$
$\displaystyle \frac{100}{85} = (1.03)^t$
$\displaystyle (1.03)^t = 1.176470588$
Now take the logarithm of both sides. I personally prefer log base e = "ln", but pick which ever base you prefer. The choice does not matter.
$\displaystyle ln \left ( (1.03)^t \right ) = ln(1.176470588)$
$\displaystyle t \cdot ln(1.03) = ln(1.176470588)$
$\displaystyle t = \frac{ln(1.176470588)}{ln(1.03)} = 5.498156798$
So t = 5.5 years or so.
-Dan