Plotting the logarithm of y against x turns an exponential relationship into a straight line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it straightens
If y = a·bˣ then log y = log a + x log b. That is linear in x, with intercept log a and slope log b.
Reading the parameters
The slope of the semi-log line is log b, so b is 10 to that slope. The intercept gives a the same way.
A model test
If a semi-log plot is straight, an exponential model fits. If it curves, the relationship is something else.
Log-log plots
Plotting log y against log x straightens a power relationship instead. Which plot straightens tells you which model applies.
Why the AP course cares
This is how exponential models are actually identified from real data, rather than assumed.
Making an exponential look straight
Plot the logarithm of the output against the input and an exponential relationship becomes a straight line. Taking logs of y = abˣ gives log y = log a + x log b, which is linear in x.
Reading the parameters off the line
The slope of the semi-log plot is log b and the intercept is log a. Undoing the logarithms recovers the growth factor and initial value, which is how exponential models are fitted to real data.
It is also a diagnostic
If a semi-log plot is straight, the data are exponential; if it curves, they are not. Linearising is a test of the model as much as a method for fitting it.
Log-log plots for power laws
Taking logs of both axes straightens a power relationship y = axⁿ instead, with the slope giving n. Which plot straightens the data tells you which family of model applies.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Marcus Read a Semi-Log Plot
A semi-log plot of some data is straight with slope 0.3 and intercept 1.
- Step 1
A straight semi-log plot means the underlying relationship is exponential.
Step 4: Your Turn
Practice makes it stick.
The Intercept
Problem 1 of 2
A semi-log plot has intercept 1, so log a = 1. What is a?
The Test
Problem 2 of 2
A semi-log plot is straight. Is an exponential model appropriate? 1 yes, 0 no.
Straighten the Data
1 of 8
log a = 2. What is a?
2 of 8
log b = 1. What is b?
3 of 8
log a = 0. What is a?
4 of 8
A semi-log plot curves rather than straightening. Is an exponential model right? 1 yes, 0 no.
5 of 8
Which plot straightens a power relationship? 1 semi-log, 2 log-log.
6 of 8
log y = 1 + 0.3x at x = 0. What is log y?
7 of 8
Match each plot with the relationship it straightens.
Tap a card on the left to start.
8 of 8
log a = 3. What is a?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A semi-log plot has intercept 2, so log a = 2. What is a?
Question 2 of 2
What does a straight semi-log plot tell you?
What You Learned
- A semi-log plot logs y and straightens an exponential relationship.
- Its slope is log b and its intercept is log a.
- A log-log plot straightens a power relationship instead.