What is the significance of ln

You cannot use the EXP function to directly unlog the error statistics of a model fitted to natural-logged data. You need to first convert the forecasts back into real units and then recalculate the errors and error statistics in real units, if it is important to have those numbers. However, the error statistics of a model fitted to natural-logged data can often be interpreted as approximate measures of percentage error, as explained below, and in situations where logging is appropriate in the first place, it is often of interest to measure and compare errors in percentage terms.

In general, the expression LOG b. In particular, LOG means base log in Excel. In the remainder of this section and elsewhere on the site , both LOG and LN will be used to refer to the natural log function, for compatibility with Statgraphics notation. 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.

This property of the natural log function implies that. Why is this important? Now observe:. For large percentage changes they begin to diverge in an asymmetric way. If you don't believe me, here's a plot of the percent change in auto sales versus the first difference of its logarithm, zooming in on the last 5 years. The blue and red lines are virtually indistinguishable except at the highest and lowest points.

If the situation is one in which the percentage changes are potentially large enough for this approximation to be inaccurate, it is better to use log units rather than percentage units, because this takes compounding into account in a systematic way, and it is symmetric in terms of sequences of gains and losses.

A diff-log of Return to top of page. Makes sense, right? If we go backwards. This means if we go back 1. Ok, how about the natural log of a negative number? Well, if we use imaginary exponentials , there is a solution. But today let's keep it real. How long does it take to grow 9x your current amount? Sure, we could just use ln 9. Any growth number, like 20, can be considered 2x growth followed by 10x growth. Or 4x growth followed by 5x growth.

Or 3x growth followed by 6. See the pattern? This relationship makes sense when you think in terms of time to grow. The net effect is the same, so the net time should be the same too and it is. How about division? In general we have. I hope the strange math of logarithms is starting to make sense: multiplication of growth becomes addition of time, division of growth becomes subtraction of time. We can consider the equation to be:. If I double the rate of growth, I halve the time needed.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. What is the reason why we use natural logarithm ln rather than log to base 10 in specifying functions in econometrics? There is no very strong reason for preferring natural logarithms. Suppose we are estimating the model:. Hence the model is equivalent to:.

For a source in an econometrics textbook saying that either form of logarithms could be used, see Gujarati, Essentials of Econometrics 3rd edition p Since you end up with exponential in the calculus, the best way to get rid of it is by using the natural logarithm and if you do the inverse operation, the natural log will give you the time needed to reach a certain growth.

Also, the good thing about logarithms be it natural or not is the fact that you can turn multiplications into additions. Basically, you need to take the limit to have an infinite number of interest rate payment, which ends up being the definition of exponential. Even thought, continuous time is not widely used in real life you pay your mortgages with monthly payments, not every seconds..

An additional reason why economists like to use regressions with logarithmic functional forms is an economic one: Coefficients can be understood as elasticities of a Cobb-Douglas function. The elasticity term is used to describe the degree of response of a change of a variable with respect to another. The only reason is that the Taylor expansion , gives an intuitive interpretation of the result. So, if you're using the log differences of GDP in the right hand side of the equation, e.

Economists like the variables that can be interpreted easily. If you plugged the different log base then the interpretability is weaker. Is this unique to economics?