Question

Please help!!!!

Answer 1

Number of trees infected after t years:

The number of trees infected after t years is given by:

$f(t) = e^{0.4t}$

Question 1:

We have to find the number of years it takes to have 21 trees infected, that is, t for which:

$f(t) = 21$

Thus:

$e^{0.4t} = 21$

To isolate t, we apply the natural logarithm to both sides of the equation, and thus:

$\ln{e^{0.4t}} = \ln{21}$

$0.4t = \ln{21}$

$t = \frac{\ln{21}}{0.4}$

$t = 7.6$

Thus, it will take 7.6 years for 21 of the trees to become infected.

Question 2:

We have to find the inverse function, that is, first we exchange y and x, then isolate x. So

$f(x) = y = e^{0.4x}$

$e^{0.4y} = x$

Again, we apply the natural logarithm to both sides of the equation, so:

$\ln{e^{0.4y}} = \ln{x}$

$0.4y = \ln{x}$

$g(x) = \frac{\ln{x}}{0.4}$

Thus, the logarithmic model is:

$g(x) = \frac{\ln{x}}{0.4}$

For an example of a problem that uses exponential functions and logarithms, you can take a look at brainly.com/question/13812761