given:
a = $800
p = $500
r = 5% = 5/500 = 0.05
n = 4 (since we are compouding quarterly)
t = ?
using formular
a = p(1 + r/n)^nt
800 = 500(1 + 0.05/4)^4xt
500(1 + 0.05/4)^4t = 800
divide both sides by 500
500(1 + 0.05/4)^4t/500 = 800/500
(1 + 0.05/4)^4t = 8/5 [if f(x) = g (x) then ln(f(x)) = lin(g(x))
ln ((1 + 0.05/4)^4t) = ln(8/5) [ apply log rule : loga(x^b) = b . loga(x)
4t ln(1 + 0.05/4) = ln(8/5)
t = ln(8/5)/4ln(4.05/4)
t = 9.45
t = 9.5 years.
