Question 967716
How long will it take to earn your first million dollars if you annually invest $1000 into an IRA at 10% interest compounded monthly. 
This answer should be calculated using logarithms so that the answer is exact.
<pre>Use Future Value of Ordinary Annuity formula: {{{FV[oa] = PMT * (((1+i/m)^(mt)-1)/(i/m)))}}}
{{{1000000 = 1000 * (((1 + .1/12)^(12t) - 1)/(.1/12)))}}}
{{{1000000/1000 = (((1 + .1/12)^(12t) - 1)/(.1/12)))}}}
{{{1000cross(1000000)/cross(1000) = (((1 + .1/12)^(12t) - 1)/(.1/12)))}}}
{{{1000 = ((((12 + .1)/12)^(12t) - 1)/(.1/12))))}}}
{{{1000 = (((12.1/12)^(12t) - 1)/(.1/12)))}}}
{{{(12.1/12)^(12t) - 1 = 100/12}}} ------ Cross-multiplying
{{{(12.1/12)^(12t) - 1 = 25/3}}}
{{{(12.1/12)^(12t) = 25/3 + 1}}} 
{{{(12.1/12)^(12t) = 25/3 + 3/3}}} 
{{{(12.1/12)^(12t) = 28/3}}} ---------- Exponential form
{{{log (12.1/12, (28/3)) = 12t}}} -------- Logarithmic form
{{{log (28/3)/log (12.1/12) = 12t}}} ---------- Applying change of base
{{{(log (28/3))/(log (12.1/12))}}}{{{"÷"}}}{{{12 = t}}}
{{{highlight_green(t = (log (28/3))/(12 * (log (12.1/12))))}}} years ------ EXACT TIME
{{{highlight_green(t = 22.42885983)}}} years ------------ APPROXIMATE TIME