Resolved: Question about the change of log(y)

Hey Erica,

Thank you for the great question!

In order to give a complete answer, I need to employ a little bit of calculus.

Suppose we have a function f(x), which bears the form given below. Let us take its derivative:

Now, let us assume that f(x) = ln(y). Using the chain rule, its derivative is going to be:

Now, we put the two derivatives equal to each other:

The numerator on the left-hand side is the relative change in y - it is relative because we take the difference between two values of y (say, y_new - y_old) and know how this difference compares to y_old. The denominator is the absolute change in x:

If we take the difference in x to be 1 (as suggested in the video), then:

If we multiply the right-hand side of this equation by 100, then b1 will be converted into percentages. Therefore, an increase of 1 unit in x will result in an increase of b1 percent in y.

To answer your second question - yes, it will make a difference whether we use natural logarithm or other bases. The reason is that the derivative of the logarithms having a basis different from e is no longer "one over the argument". However, a different basis will only scale the y-axis up or down - it won't change the overall form of the plot.

I hope this answers your question! Unfortunately, the only way I could answer it is by making use of some mathematics. But regardless, the most important formula is actually the very last one :)

Thank you and keep up the good work!

Kind regards,
365 Hristina