Analyzing Light Intensity Data using Logs

Shedding Light on the Subject: Function Models of Light Decay
Part Three - Analyzing the Data: Linearizing Data Using Logs

Visually, the difference between linear and nonlinear data is often easier to see than to tell what kind of function might fit nonlinear data. For example, in analyzing the data using difference equations, a linear relationship between the change in light intensity and the light intensity was useful to find an exponential model for the data.

In this part of the investigation, you'll explore another method of linearizing exponential data. Having more than one method provides additional evidence or support for an exponential model being used for the data.

Before continuing on, recall what is the inverse function for an exponential function. Let's start with the exponential function y = 10^x.

Suppose that we know the value of y, how can we find the value of x?

For example, if y = 100 = 102, then x = 2. The inverse function of an exponential function is called a logarithmic function and satisfies the following relationship:

log(y) = x if and only if y = 10^x.

The above equation is given for the exponential function with base 10. More generally, for an exponential function with base a:

log_a(y) = x if and only if y = a^x.

You will also need two of the basic properties of logarithms:

log(a*b) = log(a) + log(b)

log(a^x) = log(a)*x

Test Your Understanding of Logs

Use your knowledge of logarithms to answer the following items.

Give a justification for the two properties of logarithms using the fundamental relationship between exponential and logarithmic functions.
Provide justifications for each of the following steps:

y = a*b^x.

log(y) = log(a*b^x)

log(y) = log(a) + log(b^x)

log(y) = log(a) + log(b)*x
Explain why the last equation in part (2) expresses a linear relationship between log(y) and x.

In the case of the light intensity data, (d, I), a reasonable conjecture is that the light intensity is an exponential function of the depth. Therefore, a linear relationship between the log of the light intensity and the depth, (d, log(I)). linearize exponential data.

Finding a Model using Logs

Use your knowledge of logarithms to answer the following items.

Add a new column to your data set for the log base 10 of your light intensity values, log(I). Graph the log(I) versus the depth. Find a linear equation which fits this new data. Be sure to use the appropriate variables of log(I) and d in your equation.
Solve your equation for log(I) from above for light intensity I. Explain your method.
Graph the original (depth, light intensity) data with the function from item 2. How well does the model fit the data?
Examine the work that you have done. Notice that logarithms base 10 were used. What happens if you use natural logarithms, ln(x), instead of base 10 logarithms in the solution?


Make a Conjecture	Gather Data	Analyze the Data	Reflect for Illumination

Back to the beginning of this i-Math Investigation

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