# Based on the model, the solution to the equation 50=40e^( 0.027t) gives the number of years it will take for the population of country A to reach 50 million. What is the solution to the equation expressed as a logarithm?

Jan 10, 2017

Equation expressed as a logarithm is $0.027 t = \ln 50 - \ln 40$

#### Explanation:

$50 = 40 {e}^{0.027 t}$ in regard to the question indicates that the population of the country increases from $40$ million to $50$ million at a continuous rate of $0.027$ or 2.7% per annum in $t$ years.

As $50 = 40 {e}^{0.027 t}$, we have

${e}^{0.027 t} = \frac{50}{40}$

hence $0.027 t = \ln 50 - \ln 40$, when $\ln$ means Napiers logrithm i.e. to the base $e$.

This solves as follows:

$t = \frac{\ln 50 - \ln 40}{0.027} = \frac{3.9120 - 3.6889}{0.027} = 8.265$ years.