MHB Logarithmic Functions: Solving Questions & Finding Carrying Capacity

[TheFallen018]

Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to help explain both.

Here is the first of the two questions
View attachment 7620

So, part a was fairly straightforward. I calculated the differential to be (-2x+3)/x(x-1)

However, I'm not sure if I broke down the expression as much as I could have with laws of logarithms. I could only think of using the inverse of the second law of logarithms, where log(a) - log(b)= log(a/b)

Is there a way to break that up further?

As for the second question, it has to do with carrying capacity of a population model.
View attachment 7621

I got all the parts correct for this one, but I'm not sure how to get the carrying capacity of the function. I figured it to be 10,000, as that was what the numerator was. However, I'm sure that's not how it's meant to work. Despite the fact that I got the right answer, I'm not satisfied with the answer I gave.

So, how would be the correct way about solving that?

Thanks for your time. I really appreciate the help.

Kind regards,

TheFallen018

 

[MarkFL]

Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to help explain both.

Here is the first of the two questions

So, part a was fairly straightforward. I calculated the differential to be (-2x+3)/x(x-1)

However, I'm not sure if I broke down the expression as much as I could have with laws of logarithms. I could only think of using the inverse of the second law of logarithms, where log(a) - log(b)= log(a/b)

Is there a way to break that up further?

You can also apply:

$$\log_a\left(b^c\right)=c\cdot\log_a(b)$$

As for the second question, it has to do with carrying capacity of a population model.

I got all the parts correct for this one, but I'm not sure how to get the carrying capacity of the function. I figured it to be 10,000, as that was what the numerator was. However, I'm sure that's not how it's meant to work. Despite the fact that I got the right answer, I'm not satisfied with the answer I gave.

So, how would be the correct way about solving that?

Thanks for your time. I really appreciate the help.

Kind regards,

TheFallen018

To find the carrying capacity $C$, I would write:

$$C=\lim_{t\to\infty}P(t)$$

We see the numerator is constant, and the denominator goes to 1, so yes, 10,000 is correct. :)

 

[TheFallen018]

You can also apply:

$$\log_a\left(b^c\right)=c\cdot\log_a(b)$$
To find the carrying capacity $C$, I would write:

$$C=\lim_{t\to\infty}P(t)$$

We see the numerator is constant, and the denominator goes to 1, so yes, 10,000 is correct. :)

Thanks Mark, that was exactly what I was looking for. You're awesome :)