Question

Question 13

What are the two advantages of taking the natural log of a variable?

Answer 1

The natural log gives us the time needed to reach a certain level of growth.

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_e x, or sometimes, if the base e is implicit, simply log x.

e and the Natural Log are twins:

• e^x is the amount we have after starting at 1.0 and growing continuously for x units of time
• ln⁡(x) (Natural Logarithm) is the time to reach amount x, assuming we grew continuously from 1.0

e^x is a scaling factor, showing us how much growth we’d get after x units of time.

The natural log is the inverse of e, a fancy term for opposite. Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln.

Advantages

• e^x lets us plug in time and get growth.
• ln(x) lets us plug in growth and get the time it would take.

the function e^x is its own derivative, and the derivative of ln(X) is 1/X. But for purposes of business analysis, its great advantage is that small changes in the natural log of a variable are directly interpretable as percentage changes, to a very close approximation.

Because changes in the natural logarithm are (almost) equal to percentage changes in the original series, it follows that the slope of a trend line fitted to logged data is equal to the average percentage growth in the original series.

Another interesting property of the logarithm is that errors in predicting the logged series can be interpreted as approximate percentage errors in predicting the original series, even though the percentages are relative to the forecast values, not the actual values. (Normally one interprets the "percentage error" to be the error expressed as a percentage of the actual value, not the forecast value, although the statistical properties of percentage errors are usually very similar regardless of whether the percentages are calculated relative to actual values or forecasts.)

Thus, if you use least-squares estimation to fit a linear forecasting model to logged data, we are implicitly minimizing mean squared percentage error, rather than mean squared error in the original units,

And if we look at the error statistics in logged units, we can interpret them as percentages if they are not too large say, if their standard deviation is 0.1 or less.  Within this range, the standard deviation of the errors in predicting a logged series is approximately the standard deviation of the percentage errors in predicting the original series, and the mean absolute error (MAE) in predicting a logged series is approximately the mean absolute percentage error (MAPE) in predicting the original series.