Straightening Data with Logarithmic Graphs
Take logarithms of a model to make it linear: y = axⁿ gives a straight line for log y against log x, and y = abˣ gives one for log y against x. The gradient and intercept then return the constants.
What a learner can do afterwards
- Choose the axes that straighten a given model
- Recover a and n from the gradient and intercept of a log-log plot
- Say what a straight log-linear plot tells you about the data
1 · Read
Some curves turn straight when you plot their logs. If data follow y = a x^n, take logs to get log y = log a + n log x. Plot log y against log x and you get a line with gradient n and intercept log a. Plain y against x stays curved, so it cannot straighten a power law.
Read a base 10 log-log plot this way. An intercept of 2 means log a = 2, so a = 10^2 = 100, and an intercept of 3 gives a = 1000. A line through (log x = 2, log y = 1) and (log x = 4, log y = 7) rises 6 over a run of 2, so the gradient is 3 and the power is 3. A gradient of 1/2 with intercept 0 means y = sqrt(x), since a = 1.
If data follow y = a b^x, log only the y values. Logs give log y = log a + x log b, which is a line in plain x with gradient log b and intercept log a. Read the intercept first to get a, then the gradient to get b. With base 10 logs a gradient of 2 means b = 10^2 = 100.
Let the slope tell the story before you compute. A rising log-linear line means growth with b above 1, while a falling one has a negative gradient, so log b is negative and b sits below 1. Changing a only shifts the intercept, never the gradient. Keep one base through the whole question and convert back with powers of 10.
Log both axes to read powers, log only y to read exponentials, and let gradient and intercept hand back the constants.
More to read
6.5 Logarithmic Properties
Log laws are the engine of straightening. The power law log(x^n) = n log x pulls the exponent down to become a gradient. The product law log(ab) = log a + log b splits the multiplier off to become an intercept. Every straightened plot is one of these laws drawn as a line. Watch the base throughout. Base 10 logs give intercepts as powers of 10, so an intercept of 3 means a = 1000. Mixing bases mid question is the classic slip. Pick base 10, stay there, and convert back with powers of 10 at the end.
6.8 Exponential Growth and Decay
Exponential data look curved on plain axes and straight on log-linear paper. Populations, radioactive samples and compound interest all follow y = a b^x, and their log plots are lines with gradient log b. A rising line means growth with b above 1; a falling line means decay with b below 1. The sign of the slope classifies the process before you compute anything. The intercept carries the starting amount. At x = 0 the line meets log a, so a drops out at once. Then the gradient gives b.
3.9 Derivatives of Exponential and Logarithmic Functions
Logs turn multiplication into addition, which is why they linearise power laws. Differentiating log y = log a + n log x with respect to log x leaves exactly n, the gradient you measure. This is the calculus behind reading powers off slopes. The derivative viewpoint also explains why proportional errors become constant vertical scatter on a log plot. For exponentials the same idea gives constant relative growth. Differentiating log y = log a + x log b with respect to x leaves log b. Constant gradient on a log-linear plot is therefore the signature of steady percentage change.
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