Question

Ramona deposited $4,190.51 into a savings account with an interest rate of 5.2% compounded twice a year. About how long will it take for the account to be worth $9,000?

Answer 1

http://www.openbookproject.net/books/bpp4awd/_images/compound_interest.png

Answer 2

so as you can see from the above formula, what is required is to "solve" or to find "t"
so the
A or "balance" will be 9000
4,190.51 is the principal, or starting amount from
the rate is 5.2% or 5.2/100 = 0.052
the "n" period per year is 2, since it's being compounded twice a year
so now
$$
9000 = 4190.51\pmatrix{1+\cfrac{0.052}{2}}^{2t}\\
\cfrac{9000}{4190.51}=\pmatrix{1+\cfrac{0.052}{2}}^{2t}\\
2.15 = 1.026^{2t}\\
-----------------\\
\text{use log cancellation rule}\\
\color{blue}{log_{1.026}} (2.15) = \color{blue}{log_{1.026}}(1.026^{2t})\\
\color{blue}{log_{1.026}}(2.15) = 2t\\
-----------------\\
\text{using log change of base rule}\\
\cfrac{log_{10} (2.15)}{log_{10}(1.026)} = 2t
$$
now all you need to do is solve for "t" :)