Determine a Log equation in the form Y= A+BlogX for the curve of best fit for the following data.

Question

Modelling Apple Tree Growth
Year	Height of Tree ft)
1	7
2	17
3	20
4	24
5	27
6	29
7	32
8	33
9	34

Determine a logarithmic function (in the form y=a+blogx) for the curve of best fit for your data. Show all of your work and explain how you determined the best model to fit the data.

Using your table and/or scatter plot, make the following predictions and then justify your predictions algebraically using the function you developed:

  a) How tall will the tree be at 30 months?

  b) How tall will the tree be in 15 years?

  c) When will the tree reach a height of 10 feet?

  d) When will the tree reach a height of 25 feet? 

Answer 1

You can do this in your calculator.

STAT --> Edit --> then enter your data

(L₁ is for the years and L₂ is for the height).

STAT --> 9 --> ENTER

Your equation will look like: y = a + b(lnx)

Use the values they give you for a and b to get your equation.

It says ln because that is the natural log, which is log with a base of e. So if you want to write log_e instead of ln, you can do so.

Once you have your final equation, you can use it to answer the questions by plugging in the given value and solving for the correct variable.

a) x = 2.5, because 30 months is 2.5 years. Solve for y

b) x = 15. Solve for y

c) y = 10. Solve for x

d) y = 25. Solve for x

Answer 2

First create a new column of data for the "x-value". Instead of just plain numbers 1 - 9, take the log (use base 10) of each number. For instance, log(1) = 0, log(2) = 0.301, etc. It will look like this:

Then plot the log(year) on the x-axis and the height of the tree in ft on the y-axis. Then draw the best possible straight line between the data points. You'll want as many above the line as below. You'll get something like this:

Measure the slope of the line, the change in height divided by the change in log(year). This will be the "B" in the equation. Take your best guess at the y-intercept. This will be the "A" in the equation.