Question

Help would be much appreciated

Answer 1

`r(t)=(1)/(t)\ln \left((A)/(P)\right)`

Explanation:

Given function:

`A(t)=Pe^(rt)`

Divide both sides by P:

`\implies (A)/(P)=e^(rt)`

Take natural logs of both sides of the equation:

`\implies \ln \left((A)/(P)\right)=\ln e^(rt)`

`\textsf{Apply the power law}: \quad \ln x^n=n \ln x`

`\implies \ln \left((A)/(P)\right)=rt\ln e`

Apply the natural log law: ln (e) = 1

`\implies \ln \left((A)/(P)\right)=rt`

Divide both sides by t:

`\implies r=(1)/(t)\ln \left((A)/(P)\right)`