Question 155617
OK, first, you know that you have exponential growth, 
your equation looks like this,
{{{N(t)=N[0]e^((alpha*t))}}}
Use your conditions to find {{{N[0]}}} and {{{alpha}}}
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Initially, at t=0 hours, N=100
{{{N(0)=N[0]e^((alpha*0))=100}}}
{{{N[0]=100}}} since {{{e^0=1}}}
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At 5 hours, N=500,
{{{N(5)=100e^((alpha*5))=500}}}
{{{e^((alpha*5))=5}}}
{{{alpha*5=ln(5)}}}
{{{alpha=ln(5)/5}}}
{{{alpha=0.322}}}
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Put it all together,
{{{N(t)=100e^((0.322*t))}}}