Question 16752
we are given annual interest rate i = 8% or .08
the semiannual interest rate is then {{{i/2}}} = {{{.08/2}}} = .04
the per period accumulation function is then {{{(1 + .04)^n}}}
so, we have 7500 = 5000 * {{{(1 + .04)^n}}}. First divide 5000 from both sides.
1.5 = {{{(1 + .04)^n}}}
log 1.5 = n log 1.04
divide log 1.04 from both sides
n = 10.3380351 half years or 5.1690175 years.

For continous interest we need to find the force of interest (d) over the accumulation period. 
we know {{{(1 + i)^n}}} = {{{e^(nd)}}}
so {{{(1 + .08)^n}}} = {{{e^(nd)}}}
n ln 1.08 = nd  (take the natural log of both sides)
ln 1.08 = d
now the accumulation function for continuous interest is {{{e^(nd)}}}
so 5000 * {{{e^(n * ln 1.08)}}} = 7500
{{{e^(n * ln 1.08)}}} = 1.5
{{{n * ln 1.08}}} = ln 1.5 (take the natural log of both sides)
n = {{{ln 1.5/ ln 1.08}}}
n = 5.2684462

Hope this helps, let me know if you have any more financial mathematics questions.  I know them fairly well.