Logarithmic function of table

Question

How do I find the logarithmic function of a table with the below data?

Demand (D)	Price of One Shirt(P)
152	2300
159	2000
164	1700
171	1500
176	1300
180	1200
189	1000

Answer 1

You can also create a logarithmic trendline in Desmos:

1. Type the word 'table'
2. input data into the table: price into the x₁ column; demand into the x₂ column
3. Set you x-axis range to include all x₁ column values: I used 900–2400
4. Set you y-axis range to include all y₁ column values: I used 140–200
5. Type y₁ ~ a ln(bx₁) + c

About step 2

I think it makes sense to put price into the x₁ column and demand into the x₂ column because price is the independent variable and demand is the dependent variable. You also get a better fit of your data using a logarithmic trendline if you do it this way.

About step 5

After you type y₁ ~ a ln(bx₁) + c, you will get an output with four numbers:

1. the coefficients a, b, and c for the equation y = a ln(bx) + c that best fits the data
2. an R² term that indicates how good the fit is (the closer to 1, the better).

You will also see a trendline drawn on your graph as a visual depiction of the curve fit you've done.

(BTW, you don't actually need the parenthesis around bx₁ for your input equation.)

Answer 2

Instead of using online curve fitting aids, you can just use simple linear regression, which is probably explained in your text book.

Letting x be the demand and y be the price, you can use the model:

y = aln(bx)+c

y = a(ln(b)+ln(x))+c

y = aln(b)+aln(x)+c

aln(b) and c can be combined into a single constant, y₀, and 'a' becomes the slope m.

y = m ln(x) + y₀

If you replace each of the x values with X=ln(x) you have

y = mX+y₀

Which is a linear equation. Using the linear regression equations, you get

y-hat = -6146.29526X+33122.11822

or

y-hat = -6146.29526ln(x)+33122.11822

Therefore the magic of finding the logarithmic equation is better understood.

You can also get a pretty good fit by taking the first and last points and solve for m and b.

2300 = ln(152)m+b

1000 = ln(189)m+b

m=-6018.5185

b = 32537.037

which is close to what the linear regression determined.

Answer 3

Load the data using STAT EDIT on a TI 83 or 84.

X values go in column L1. Y values go in column L2.

Then use STAT CALC and pick the log option to fit the data and give you the equation.

Let me know if you have any more questions.

You can also use the given function in Y= to graph the function. You can also turn on the data-point graph to see the data points and see how good the fit is. Also there is an option to display the correlation coefficient, which is a good measure of "goodness of fit".

NOTE: You can probably use a TI calculator on a test (including STAAR, SAT, & ACT). I doubt you can use a laptop in these cases. If you use Casio, let me know.

Answer 4

This requires using python or excel. Manually is too much work.

WITH EXCEL:

We will use excel’ s functionality for creating scatter plots with logarithmic trendlines. Here’s how you can do it step-by-step. For excel, assume that your data is placed in columns A (Demand) and B (Price) starting from the second row. The first row is used for headers, "A1" is "Demand (D)" and "B1" is "Price of One Shirt(P)".

Step 1: Enter the Data

1. A1: Demand (D)
2. B1: Price of One Shirt(P)
3. A2 to A8: 152, 159, 164, 171, 176, 180, 189 (Enter these demands sequentially from A2 to A8)
4. B2 to B8: 2300, 2000, 1700, 1500, 1300, 1200, 1000 (Enter these prices sequentially from B2 to B8)

Step 2: Create a Scatter Plot

1. Highlight the range A1:B8.
2. Go to the "Insert" tab.
3. In the Charts group, click on the "Scatter" chart type.
4. Choose the "Scatter with only markers" chart type.

Step 3: Add a Logarithmic Trendline

1. Click on one of the data points in your scatter plot. All points should become selected.
2. Go to the "Chart Tools Design" tab.
3. Click "Add Chart Element" > "Trendline" > "More Trendline Options".
4. In the "Format Trendline" pane, select "Logarithmic".
5. Check the "Display Equation on chart" and "Display R-squared value on chart" (if desired) to show the regression equation and R² value.

Step 4: Using the Equation

1. Once you’ve added the logarithmic trendline and displayed the equation, you should see an equation on your chart something like: y = a⋅ln(x)+b

Now you can use this function to calculate estimated price values for different demand values. To apply this manually in Excel:

1. Note down the a and b values from the chart.
2. Use the formula in another cell. Suppose you want to find the estimated price for a demand of 165: Estimated Price=a⋅ln(165)+b
3. Replace a and b with the noted values from the equation on the chart.

Example Formula Application:

Assume that the function from the trendline is: y = 500⋅ln(x)−1000

In a cell, you could write a formula like this to calculate the price for a demand of 165:

= 500 * LN(165) - 1000

Replace the coefficient and constant with those from your actual trendline equation. Remember that these steps provide a mathematical model to fit your data and predict future points, but it’s based on the assumption that future data will continue to follow the same trend. Always apply caution when making predictions with models!

WITH PYTHON:

import numpy as np

import matplotlib.pyplot as plt

from scipy.optimize import curve_fit

# Given data

D = np.array([152, 159, 164, 171, 176, 180, 189])

P = np.array([2300, 2000, 1700, 1500, 1300, 1200, 1000])

# Define the form of the function you want to fit.

# a * np.log(b * D) can be a good starting point.

def func(D, a, b):

return a * np.log(b * D)

# Use curve_fit to fit the function to the data.

popt, pcov = curve_fit(func, D, P)

# Now, popt contains the optimized parameters (a and b).

a, b = popt

# Generate x values

D_fit = np.linspace(min(D), max(D), 100)

# Generate fitted y values

P_fit = func(D_fit, a, b)

# Plot the original data as scatter

plt.scatter(D, P, label='Original Data')

# Plot the fitted curve

plt.plot(D_fit, P_fit, 'r', label=f'Fitted Curve: {a:.2f} * log({b:.2f} * D)')

plt.xlabel('Demand (D)')

plt.ylabel('Price of One Shirt(P)')

plt.legend()

plt.title('Logarithmic Fit Example')

plt.show()

In the script above:

1. D and P are your original data points.
2. func(D, a, b) defines the logarithmic function to fit the data.
3. curve_fit(func, D, P) will try to find the best parameters a and b to fit your data.
4. The fitted parameters a and b are then used to create a fitted curve which is plotted alongside your original data points. 

The form of the function a * np.log(b * D) might need to be adjusted depending on the exact nature and behavior of your data.