Question

Logarithms and stuff, please help

Answer 1

Answer:

$\displaystyle t=-25\log\left(\frac{20}{30}\right)$

Step-by-step explanation:

Logarithms

The model that predicts the percentage of voters (V) as a function of the number of years (t) is:

$V(t)=100-30\cdot e^{-0.04t}$

It's required to find in how many years the value of V will be V=80%. The equation to solve is:

$100-30\cdot e^{-0.04t}=80$

Subtracting 100:

$-30\cdot e^{-0.04t}=80-100=-20$

Dividing by -30:

$\displaystyle e^{-0.04t}=\frac{20}{30}$

Applying logs:

$\displaystyle \log( e^{-0.04t})=\log\left(\frac{20}{30}\right)$

For logs property:

$\displaystyle -0.04t\log(e)=\log\left(\frac{20}{30}\right)$

log(e)=1. Solving for t:

$\displaystyle t=\frac{\log\left(\frac{20}{30}\right)}{-0.04}$

Since 0.04=1/25:

$\boxed{\displaystyle t=-25\log\left(\frac{20}{30}\right)}$