Logarithmic data transformations

Logarithmic explanatory coordinates

Use the original positive x-values to calculate their base-10 logarithms. The output y is unchanged. Equal multiplicative x steps become equal additive steps in t

1. 1Log the explanatory variable only For the General logarithmic input model, plot y against t=log₁₀(x). Keep the response on its original scale; logging y would test a different model
2. 2Require positive inputs All original inputs must be strictly positive. A negative transformed input is allowed when 0<x<1.
3. 3Check every transformed slope Compute with the transformed horizontal differences. Powers of ten give exact integer inputs; for other values retain calculator precision. Unequal slopes reject exact linearity

Data or rule

\(\displaystyle x=0.1,\quad t=\log_{10}(x)\)

Transform the positive input

Data or rule

\(\displaystyle x=1,\quad t=\log_{10}(x)\)

Transform the input corresponding to the transformed vertical axis

Which list gives the transformed t and y values for (x,y)=(1,7),(10,10),(100,13), where t=log₁₀(x)?

Data or rule

\(\displaystyle (t,y)=(0,7),(1,10),(2,13)\)

Calculate the slope between the last two transformed observations

All four original observations

\(\displaystyle (x,y)=(1,7),(10,10),(100,13),(1000,17)\)

After adding the fourth observation, does plotting y against log₁₀(x) still give an exact line?

Move through the logarithmic input coordinates. Observe the original input and unchanged response. Each increase of one in t multiplies x by ten

At the transformed-line intercept t=0, what is the original input?

Test the logarithmic transformation independently

Use the exact observations (x,y)=(0.1,2),(1,5),(10,8),(100,11). Produce the transformed table and scatterplot, label the axes and choose a useful window. Verify every consecutive slope. Then change only the final response to 12 and explain whether the same transformation remains exactly linear. State the original-input domain and explain why this is not a transformation of the response

1. 1Write before continuing Produce your complete table, diagram or explanation independently. The next screen is a model for self-review; the complete written response is not automatically marked

Model response: Test the logarithmic transformation independently

Compare your working, labelled axes, numerical checks and conclusion with this model

1. 1Table and plot The transformed points are (−1,2),(0,5),(1,8),(2,11). Use horizontal axis t=log₁₀(x), vertical y and, for example, −1.5≤t≤2.5, 0≤y≤13. All three consecutive slopes are three
2. 2Counterexample Changing the last response to 12 makes the last slope four, while the other two remain three. The modified exact observations are not collinear after logging x.
3. 3Interpretation Original x must be positive; negative t is permitted. The response values were retained unchanged. Agreement at four exact observations supports this candidate for the supplied dataset but does not establish a universal practical law