Logarithms for Skew and Growth

Most Pantry baskets cost between ₹300 and ₹800. A handful of office pantries order ₹40,000 of snacks at once. Plot order values on an ordinary axis and you get one tall spike near zero and a long, empty tail. Plot the monthly orders of a fast-growing new delivery zone and the early months look flat. Both pictures are fixed by the same tool.

The rule

Logs turn multiplying into adding: equal ratios become equal steps.

The base-10 logarithm asks "10 to what power gives this number?"

Each step of ×10 in value is a step of +1 in the log. On a log axis, ₹400 and ₹4,000 sit as far apart as ₹4,000 and ₹40,000, so both ordinary baskets and office orders become visible on one chart.

Three reasons data scientists reach for logs:

• Skewed data such as order values, incomes and page views spreads out evenly.
• Growth at a steady percentage rate draws a straight line on a log scale, so a bend means the growth rate itself changed.
• Ratios become differences: $\log(a/b) = \log a - \log b$. Doubling is always +0.301 in log10 and halving is always −0.301, which makes rises and falls symmetric, unlike percentages.

The natural log (written ln, base $e$) has a handy extra: for small changes, a rise of 0.05 in ln is close to a 5% rise in the value. That is why models often predict the log of spend, and why their coefficients read as percentages.

One catch: the log of zero does not exist. For a column with zeros, such as spend in a week with no orders, use $\log(1 + x)$, which keeps zeros at zero.

The shortcut

Count the zeros, then add the log of what is left. For 4,000: three zeros give 3, and log10(4) is about 0.6, so the answer is about 3.6. To compare two values, subtract their logs, because a difference of logs is the log of their ratio.

Drills

Answer fast — trust the rule.

Remember

• Logs turn ratios into steps: each ×10 is +1 in log10.
• Count the zeros, add the log of the rest; subtract logs to compare values.
• Trap: the log of zero. Use log(1 + x) for columns that contain zeros.