# Relationship between log and natural

### Exponential and logarthmic functions | Khan Academy

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 ♤ The natural logarithm of x is generally written as ln x, loge x, or sometimes, . The connection between area and the arcs of circular and hyperbolic. This property of the natural log function in auto sales versus the first difference of its logarithm. Natural Log (ln) is the amount of time needed to reach a certain level of . there isn't much difference between yearly compounded and fully continuous interest.

Sure, we could just use ln 4. We can consider 4x growth as doubling taking ln 2 units of time and then doubling again taking another ln 2 units of time: 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. This relationship makes sense when you think in terms of time to grow. If we want to grow 30x, we can wait ln 30 all at once, or simply wait ln 3to triple, then wait ln 10to grow 10x again. The net effect is the same, so the net time should be the same too and it is.

Well, growing 5 times is ln 5. Suppose we want 30x growth: We can consider the equation to be: If I double the rate of growth, I halve the time needed. The natural log can be used with any interest rate or time as long as their product is the same. To demonstrate this point, here's a graph of the first difference of logged auto sales, with and without deflation: By logging rather than deflating, you avoid the need to incorporate an explicit forecast of future inflation into the model: Logging the data before fitting a random walk model yields a so-called geometric random walk --i.

A geometric random walk is the default forecasting model that is commonly used for stock price data. 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. It is much easier to estimate this trend from the logged graph than from the original unlogged one! If we had instead eyeballed a trend line on a plot of logged deflated sales, i.

Usually the trend is estimated more precisely by fitting a statistical model that explicitly includes a local or global trend parameter, such as a linear trend or random-walk-with-drift or linear exponential smoothing model.

### Uses of the logarithm transformation in regression and forecasting

When a model of this kind is fitted in conjunction with a log transformation, its trend parameter can be interpreted as a percentage growth rate.

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, albeit 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, you are implicitly minimizing mean squared percentage error, rather than mean squared error in the original units, which is probably a good thing if the log transformation was appropriate in the first place.

And if you look at the error statistics in logged units, you can interpret them as percentages if they are not too large--say, if their standard deviation is 0. 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.

I am using a benchmark of 0. But the original equation says that that exponent is x. Note that if c is negative then there is no real solution. However there is a complex solution. The base 10 logarithm function is defined to do exactly the opposite, namely: Therefore these are inverse functions. We should look at these two equations as expressing the same relationship between x and y but from different points of view.

The first equation is the relationship solved for y and the second one is the relationship solved for x. If we replace x in the first equation by the x of the second equation we get this identity: This always happens with inverse functions.

How to use the base 10 logarithm function in the Algebra Coach Type log x into the textbox, where x is the argument.

The argument must be enclosed in brackets. Set the relevant options: In floating point mode the base 10 logarithm of any number is evaluated. In exact mode the base 10 logarithm of an integer is not evaluated because doing so would result in an approximate number. Turn on complex numbers if you want to be able to evaluate the base 10 logarithm of a negative or complex number. Click the Simplify button.

- Exponentials & logarithms
- Demystifying the Natural Logarithm (ln)

Algorithm for the base 10 logarithm function Click here to see the algorithm that computers use to evaluate the base 10 logarithm function. The natural logarithm function Background: You might find it useful to read the previous section on the base 10 logarithm function before reading this section.

## Natural logarithm

The two sections closely parallel each other. But why use base 10?

After all, probably the only reason that the number 10 is important to humans is that they have 10 fingers with which they first learned to count. Maybe on some other planet populated by 8-fingered beings they use base 8!