A.3 log10 transformations

At its simplest, log10 transformations return base 10 logarithms. For example, since \(1000 = 10^3\), running log10(1000) returns 3 in R. To undo a log10 transformation, we raise 10 to this value. For example, to undo the previous log10 transformation and return the original value of 1000, we raise 10 to the power of 3 by running 10^(3) = 1000 in R.

Log transformations allow us to focus on changes in orders of magnitude. In other words, they allow us to focus on multiplicative changes instead of additive ones. Let’s illustrate this idea in Table A.1 with examples of prices of consumer goods in 2019 US dollars.

TABLE A.1: log10 transformed prices, orders of magnitude, and examples
Price log10(Price) Order of magnitude Examples
$1 0 Singles Cups of coffee
$10 1 Tens Books
$100 2 Hundreds Mobile phones
$1,000 3 Thousands High definition TVs
$10,000 4 Tens of thousands Cars
$100,000 5 Hundreds of thousands Luxury cars and houses
$1,000,000 6 Millions Luxury houses

Let’s make some remarks about log10 transformations based on Table A.1:

  1. When purchasing a cup of coffee, we tend to think of prices ranging in single dollars, such as $2 or $3. However, when purchasing a mobile phone, we don’t tend to think of their prices in units of single dollars such as $313 or $727. Instead, we tend to think of their prices in units of hundreds of dollars like $300 or $700. Thus, cups of coffee and mobile phones are of different orders of magnitude in price.
  2. Let’s say we want to know the log10 transformed value of $76. This would be hard to compute exactly without a calculator. However, since $76 is between $10 and $100 and since log10(10) = 1 and log10(100) = 2, we know log10(76) will be between 1 and 2. In fact, log10(76) is 1.880814.
  3. log10 transformations are monotonic, meaning they preserve orders. So if Price A is lower than Price B, then log10(Price A) will also be lower than log10(Price B).
  4. Most importantly, increments of one in log10-scale correspond to relative multiplicative changes in the original scale and not absolute additive changes. For example, increasing a log10(Price) from 3 to 4 corresponds to a multiplicative increase by a factor of 10: $100 to $1000.