Edward is collecting data on the average length of lizards from head to tail for a particular generation. Initially, the average length of the lizards was 20 centimeters, and the lizards grew at a relatively fast rate. Two years later, the average length of the lizards was about 22.197 centimeters. With each passing year, the lizards still increased in length, but the growth rate slowed considerably. Which equation in the general form of a natural logarithmic function, , represents the average length of lizards in this generation? (Hint: The graph passes through the point (h + 1,k).)

The general form of a natural logarithmic function is f(x)=a In(x-h) +k. Since the initial average length of the lizards was 20 centimeters, one possible point (h+1,k) through which the function passes is (0,20).

In this case, h= -1 since h +1 = 0 and k = 20.

It follows that x - h = (-1) = x + 1.

So, the function takes the form f(x) = a In(x + 1) +20.

The average length of the lizards after 2 years is 22.197. So f(2) =22.197. We substitute this value into the function

Step-by-step explanation:

f(2) = a In(2 +1) +20

22.197 = a In(3) + 20

[tex]\frac{2.197}{1.099}[/tex]≈ a

2 ≈ a

So the function that represents the average length of the lizards across this generation is f(x) = 2 In(x+1) +20