Question

Logarithms and stuff, please help

Answer 1

`\displaystyle t=-25\log\left((20)/(30)\right)`

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)=(20)/(30)`

Applying logs:

`\displaystyle \log( e^(-0.04t))=\log\left((20)/(30)\right)`

For logs property:

`\displaystyle -0.04t\log(e)=\log\left((20)/(30)\right)`

log(e)=1. Solving for t:

`\displaystyle t=(\log\left((20)/(30)\right))/(-0.04)`

Since 0.04=1/25:

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