Berkeley purchased a small painting for $5,000 and she was told it should increase in value approximately 10% per year. how long will it take the painting to reach a value of $20,886.24? (round to the nearest year)

This is exponential growth of the form:

f=ir^t, f=final value, i=initial value, r=common ratio, t=time, in this case:

f=5000((100+10)/100)^t

f=5000(1.1)^t

And we wish to know when f=20,886.24 so

20886.24=5000(1.1)^t  divide both sides by 5000

4.177248=1.1^t  take the natural log of both sides

t=ln4.177248/ln1.1

t≈14.999999574476086460115187057714

t≈15 years (to nearest year)