Question 812083
how many years will it take $5,500 to grow to $6,300 if it is invested at an annual rate of 2.5% compounded continuously?
<pre>
A = Pe<sup>rt</sup>

A = final amount = 6300 
P = beginning amount = 5500
r = rate expressed as a decimal = 0.025
t = time in years = (the unknown)

               A = Pe<sup>rt</sup> 
  
Solve for t. Divide both sides by P

              {{{A/P}}} = e<sup>rt</sup>  

Write an equivalent natural log equation using the rule that 
X = e<sup>Y</sup> is equivalent to Y = ln(X)

              rt = ln{{{(A/P)}}}

Use the rule of logs ln{{{(X/Y)}}} = ln(X)-ln(Y)

              rt = ln(A)-ln(P)

Divide both sides by r

               t = {{{(ln(A)-ln(P))/r}}}

Now substitute the given values 

               t = {{{(ln(6300)-ln(5500))/0.025}}}     

Put that in your TI-83 or TI-84 this way

(ln(6300)-ln(5500))/0.025

Press ENTER

Read 5.432061646 years or about 5 years and 5 months.

Edwin</pre>