So, uh, in this case, we're going to use the natural growth model. So the formula for that is p of T equals peanut times e to the rt where yeah, p of t is the and population p not is the starting population are is the rate of growth. We're decline and T is the time. In this case, if we're going to predict how long it takes for the population to double, we need to know the rate at which the population is growing. To do that, we can use the example given in this case, the population of the otters went from 500 to 604 years. So that means, in our case, we have pe four equals 600 when p not was 500. So this means we have values for three of the four variables and so therefore we can get a value for are so we substitute these values into the formats of 600 equals 500 times e to the R times four and we solve for R. So divide both sides by 500. So that's 6/5 equals e to the four are then we can use laws of logarithms to rewrite this exponential equation as a natural algorithm. So the natural log of 6/5 equals four R and then divide both sides by four. So we get the natural log of 6/5 over. Four eagles are so now. If we want to look at how long it takes the population to double, we can use this value for our and we want our population to double so we could use the same starting population of 500. Or if we choose, we could just let the original population B one, which means our P value it would be, too, because it's going to double. And so we plug in those values and this time we're going to solve for T. So we've got two equals one time, see to the natural log of 6/5 over four times t. So that's one times Anything is that other objects it would just be too equals e to the natural log of 6/5 over 14. Use rules of algorithms to once again change this into a natural log equation. So the natural log of two equals the natural log of 6/5 over four times t multiplied by the both sides by the reciprocal mhm. And so we get t equals four times the natural log of two over the natural log of 6/5. And we can use a calculator to get an approximate answer for this problem. So when I put this into my calculator, I get four times. The natural log of two, divided by the natural log of six ifs, is approximately 15.2 years.