Logarithmic Scales: Seeing Growth Clearly

Why charts of long-run growth, stock markets and pandemics often use logarithmic scales, and how to read them without being misled.

Look at a chart of the Sensex since 1980 on a normal scale, and the early decades look almost flat, with a steep rise at the end. Look at it on a logarithmic scale, and a different picture appears. Knowing the difference is essential for reading economic charts.

Linear scales

On a linear scale, equal distances on the axis represent equal amounts. The gap from 100 to 200 is the same as from 10,000 to 10,100.

Logarithmic scales

On a logarithmic or log scale, equal distances represent equal percentage changes or multiples. The gap from 100 to 200 (doubling) is the same as from 10,000 to 20,000 (also doubling).

Why use log scales

Growth over long periods: when something grows at a steady percentage rate, it rises faster and faster on a linear chart. On a log chart, steady growth appears as a straight line.
Comparing periods: a 10 percent fall in the 1990s and a 10 percent fall today look the same size on a log chart.
Wide ranges: data spanning many sizes, such as GDP of small and large countries, fit better.

Examples

Stock indices over decades.
GDP per capita over centuries.
COVID-19 cases early in the pandemic, when growth was exponential.
Moore’s law: chip transistor counts rising exponentially.

Reading log charts

Check the axis labels: log scales often go 1, 10, 100, 1,000.
A straight line means a constant growth rate.
A steeper line means faster percentage growth.
A flattening line means growth is slowing, even if absolute numbers are still rising.

How charts can mislead

A linear chart can make recent changes look dramatic and early ones invisible.
A log chart can make large absolute changes look small.
Neither is wrong; they answer different questions. Always check which scale is used.

Two views of the Sensex

On a linear chart, the Sensex's rise from 1,000 to 2,000 in the early 1990s is barely visible, while a fall from 80,000 to 72,000 looks huge. On a log chart, the doubling in the 1990s appears much larger than the recent 10 percent fall, because it was a far bigger percentage change.

Thinking a straight line on a log chart means no growth

A straight upward line on a log chart means steady percentage growth, which is exponential growth in absolute terms.

Key takeaways

Linear scales show equal amounts; log scales show equal percentage changes.
Steady percentage growth appears as a straight line on a log scale.
Log scales suit long-run growth, stock indices and exponential processes.
Always check the scale, since each can mislead in different ways.