Answer:
[tex]\displaystyle t=-25\log\left(\frac{20}{30}\right)[/tex]
Step-by-step explanation:
Logarithms
The model that predicts the percentage of voters (V) as a function of the number of years (t) is:
[tex]V(t)=100-30\cdot e^{-0.04t}[/tex]
It's required to find in how many years the value of V will be V=80%. The equation to solve is:
[tex]100-30\cdot e^{-0.04t}=80[/tex]
Subtracting 100:
[tex]-30\cdot e^{-0.04t}=80-100=-20[/tex]
Dividing by -30:
[tex]\displaystyle e^{-0.04t}=\frac{20}{30}[/tex]
Applying logs:
[tex]\displaystyle \log( e^{-0.04t})=\log\left(\frac{20}{30}\right)[/tex]
For logs property:
[tex]\displaystyle -0.04t\log(e)=\log\left(\frac{20}{30}\right)[/tex]
log(e)=1. Solving for t:
[tex]\displaystyle t=\frac{\log\left(\frac{20}{30}\right)}{-0.04}[/tex]
Since 0.04=1/25:
[tex]\boxed{\displaystyle t=-25\log\left(\frac{20}{30}\right)}[/tex]
