Log vs Linear Scale

There are two common kinds of charts used when presenting a graph of something like a stock price over time. One is linear and the other logarithmic.

Remember that with most such charts, the "X axis" (the horizontal measure) reflects time passed. So it might show you days, or weeks, or months, or years. The "Y axis" (the vertical measure) shows you how the stock price has changed over the given time period. Linear and logarithmic charts differ in their treatment of the Y axis. We'll first explain how the charts differ in their structure, and then why it's good to use log charts.

Linear: With a linear chart, the Y axis is structured in such a way that an equal distance along the axis represents an equal absolute change in stock price. (Gee, that wasn't too clear, was it? Let's try an example.) Shifting up three spaces on the vertical axis might represent a change in stock price from $10 to $13. Shifting up another three spaces further up on the axis might represent a change in stock price from $45 to $48. Three spaces, three dollars. Every time.

Logarithmic: With a logarithmic chart, the Y axis is structured in such a way that an equal distance along it represents an equal *percentage* change. So if you move up three spaces on the vertical axis, that might represent an increase of 15% in the stock price (perhaps from $20 to $23). Shifting up another three spaces further up on the axis will represent another 15% change in the stock price, but this time perhaps from $80 to $92. Note that in the first case, the absolute price change was $3. But further up the axis, the three spaces represented an absolute price change of $12.

An example of a linear chart.

An example of a logarithmic chart.

Now -- why is one kind of chart better than another? Well, imagine a company with a stock price increasing by 15% each year for 20 years. Think about how you'd normally draw a chart of its stock price. You'd probably use a linear chart, as that's what most of us learned to do in school. The graph would show a really curved line, though. It would look like the stock price grew slowly in the first years, and then zoomed up a lot in the last few years.

That's because in the first few years, the change in the stock's price might have been from $10 to $11.50, and later from $25 to $28.75, and later still from $75 to $86.25. So the absolute changes will look small at the beginning, and will look large later on. But it's really just been a steady 15% increase from year to year. (Remember, an investor should be just as happy with a total 50% return from $20 to $30 as from $100 to $150. Investment-wise, percentage-wise, it's the same thing.)

This is why a logarithmic chart is preferable in this situation. If a company is growing at a steady clip, you'll see a fairly straight, upward-sloping line on the graph, not a sharply curving line. If the company's growth is slowing, you'll see the upward slope taper off a bit. If the company keeps growing faster and faster, then you'll see an upward-sloping sharp curve.

Courtesy Fool.com