IB Maths AI HLNon-linear RegressionPaper 1 & 2~7 min read
Logarithmic Scales
Some data spans a huge range — country populations run from a few hundred to over a billion. A normal axis can’t show that without cramming most values into a tiny strip. A logarithmic scale fixes this: instead of going 1, 2, 3, 4… it goes 1, 10, 100, 1000… so each step multiplies. Swap the variable for its logarithm and those exploding intervals become evenly spaced. The payoff: power curves straighten out on log-log graphs, and exponential curves straighten out on semi-log graphs — far easier to read.
📘 What you need to know
A logarithmic scale increases by multiplication (1, 10, 100, 1000…) instead of addition (1, 2, 3, 4…).
Replacing a variable with its log turns an exponential-looking scale into evenly spaced steps (1, 2, 3, 4…).
Why use it: to display data with a very large range, or to study rate of growth rather than actual values.
Log-log graph: both axes logged (log x and log y). Power functions y = axb become straight lines.
Semi-log graph: only the y-axis logged (log y). Exponential functions y = abx become straight lines.
Any base works — usually 10 (log) or e (ln). Watch which one a question uses.
To get back a value from a log: log x = k ⇒ x = 10k; ln x = k ⇒ x = ek.
Normal scale vs log scale
On a normal scale, equal gaps mean equal differences. On a log scale, equal gaps mean equal multiples (×10 each step).
linear scale vs logarithmic scale
Equal spacing on a log scale means equal multiples — so values from 1 to 1000 fit in the same room as 0 to 1000 on a linear scale.
The trick: if x runs 1, 10, 100, 1000… use log x and it becomes 0, 1, 2, 3…. If x runs 1, e, e², e³… use ln x instead.
Log-log vs semi-log graphs
Which graph straightens which curve is the key fact — and it’s exactly what the exam tests.
Log-log graph
power → line
BOTH axes logged (log x and log y). A power function y = axb becomes a straight line.
Semi-log graph
exponential → line
ONLY the y-axis logged (log y). An exponential function y = abx becomes a straight line.
🧠 Memory aid — “two logs for power, one for exponential”
Power needs both variables logged (log-log) to straighten — because the variable x is the thing being raised to a power. Exponential only needs the y-axis logged (semi-log) — because x is already in the exponent. So: power = log-log, exponential = semi-log.
Reading values off a log graph
When you read a value off a log axis, you’re reading a logarithm — so undo it to get the real value.
Undoing a logarithm
log x = k ⇒ x = 10k ln x = k ⇒ x = ekcheck the base: log → power of 10, ln → power of e
🧭 Recipe — estimating a value from a log-log / semi-log graph
Check the axes: is it log x, log y, or both? Note the base (log or ln).
Convert your given value to its logarithm (e.g. x = 56 → log 56 = 1.748).
Read across/up the graph to find the matching logged value on the other axis.
Undo the logarithm to get the actual value (10k or ek).
Worked examples
WE 1
Which graph straightens which curve?
State the type of graph on which each becomes a straight line. (a) y = 3x2.5. (b) y = 4(1.8)x.
(a) y = 3x^2.5 is a power function(a) log-log graph(b) y = 4(1.8)^x is exponential(b) semi-log graphpower → log-log; exponential → semi-log.
WE 2
Pick the right variable
An x-axis shows values 1, e, e², e³, … To get evenly spaced steps, which variable should you plot?
the values multiply by e each steptaking ln turns 1, e, e², e³ into 0, 1, 2, 3.plot ln xmatch the log base to the scale: powers of 10 → log, powers of e → ln.
WE 3
Estimate from a log-log graph
A function y = f(x) is drawn on a log-log graph (axes log x and log y, base 10). Show that when x = 56, y ≈ 24. (Reading the graph at log x = 1.748 gives log y ≈ 1.375.)
Step 1: log xx = 56 → log 56 = 1.748…Step 2: read off log y from the graphlog y ≈ 1.375Step 3: undo the logy = 10^1.375 = 23.71…y ≈ 24 ✓log the input, read off, then raise 10 to the power.
WE 4
Undo a natural log reading
On a semi-log graph the y-axis is ln y. At a certain point ln y = 2.3. Find the actual value of y.
ln y = k ⇒ y = e^ky = e^2.3y ≈ 9.97base e → raise e to the power (not 10).
WE 5
Why a log scale here?
Country populations range from about 800 to 1 450 000 000. Why is a logarithmic scale better than a normal one for plotting these?
the range is enormouson a normal scale, all the small countries would be squashed into a tiny strip near zero.a log scale spreads the values out evenlylog scales suit data spanning many orders of magnitude.
💡 Top tips
Power → log-log; exponential → semi-log. Memorise this pairing.
Check the base before undoing a log: log → 10k, ln → ek.
Match the variable to the scale: powers of 10 → log x; powers of e → ln x.
A value read off a log axis is a logarithm — remember to convert it back.
Use a log scale for huge ranges or when you care about rate of growth.
Show your working: log the input, read off, then raise 10 (or e) to the power.
⚠ Common mistakes
Mixing up log-log and semi-log. Power = both axes; exponential = y-axis only.
Forgetting to undo the log. A reading of 1.375 isn’t y — y = 101.375.
Using the wrong base. ln readings need ek, not 10k.
Treating the log axis as linear. Steps multiply, they don’t add.
Plotting log x when the scale is powers of e — use ln x instead.
Reading values without checking which axes are logged.
Next up — Linearising using Logarithms. You’ve seen that logs straighten power and exponential curves; next you’ll use that to actually find the model. By taking logs of y = abx or y = axb, you turn it into a straight line Y = mX + c, run a linear regression, then recover a and b.
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