exponential and logarithmic models

Question

Find how many years it would take for an investment of 3800 to grow 9000 at annual interest rate of 5.7% compounded quarterly.

Answer 1

First we need to set up an equation using the compound interest formula A=P(1+r/n)^nt where

A is the final amount in this case that would be 9000

P is principal in this case 3800

r is the rate as a decimal point .057

n is the number of times per year it compounds for quarterly n=4

t is the time in years which is what needs to be found So the equation with every value entered will be 9000=3800(1+.057/4)^4t

First divide both sides by 3800

9000/3800=(1+.057/4)^4t Reduce the fraction

45/19=(1+.057/4)^4t

45/19=(1.01425)^4t Take the ln of both sides ln(45/19)=ln(1.01425)^4t Move the 4t in front ln(45/19)=4tln(1.01425) Divide both sides by ln(1.01425)

ln(45/19)/ln(1.01425)=4t Divide both sides by 4 ln(45/19)/4ln(1.01425)=t

t=15.23425 years~15 years