Answer:
[tex]5.5\ years[/tex]
Step-by-step explanation:
we know that
In this problem we have a exponential function of the form
[tex]f(x)=a(b^{x})[/tex]
where
x is the time in years
f(x) is the value of the stock
a is the initial value
b is the base
r is the rate
b=(1-r)
we have
[tex]a=\$15,000[/tex]
[tex]r=4\%=4/100=0.04[/tex]
[tex]b=(1-0.04)=0.96[/tex]
substitute
[tex]f(x)=15,000(0.96^{x})[/tex]
80% of original price is equal to
[tex]f(x)=0.80(15,000)=12,000[/tex]
so
For f(x)=12,000 ------> Find the value of x
[tex]12,000=15,000(0.96^{x})[/tex]
[tex](12/15)=(0.96^{x})[/tex]
Apply log both sides
[tex]log(12/15)=log(0.96^{x})[/tex]
[tex]log(12/15)=(x)log(0.96)[/tex]
[tex]x=log(12/15)/log(0.96)[/tex]
[tex]x=5.5\ years[/tex]
