A Simple Discussion of the Very Useful ---- ln(X) ---- Function
Economists make extensive use of the ln(X) function, known as the natural log function.
We often write it as log(X), although strictly speaking log(X) is the common, rather than
the natural, logarithmic function. Economists nearly always use the natural logarithmic
function, so we freely interchange ln(X) and log(X), withut worrying about the base. The
base is nearly always assumed to be Napier’s constant e, roughly 2.71828.
Let’s try and discuss the log function as simply as possible, without getting involved in
complicated proofs. Naturally, we will want to drive all of the important rules regarding
the use of the ln(X) function.
Definition: ln(X) is equal to the area in the graph below.
The same is true if X is less than 1, except that the gray area now becomes a negative
number. Therefore, we have immediately that ln(1) = 0, ln(X) > 0, if X >1, and
ln(X) < 0, if X < 1. The function ln(X) should be thought of as measuring the gray area
in the graph above. The gray area in the graph above will be equal to 1 when X = e (or
roughly 2.71828).
Next, suppose that X increases to X + ΔX. How much does ln(X) change?
To see this we note approximately how much the gray area in the graph above changes
when X increases by ΔX. Clearly, the black area in the graph below is approximately
equal to Δln(X) ≈ ΔX/X which is called the growth rate of X by economists, when the
change occurs over a fixed time interval. This relation is so important that we state it
again formally.
Δln(X) ≈ ΔX/X for small changes ΔX.
This is especially useful for calculating approximate growth rates and inflation rates. If
Yt = GDP in year t, then the growth of Yt for year t is approximately equal to
ln(Yt) – ln(Yt-1). If πt is the inflation rate for year t, then πt ≈ ln(Pt) – ln(Pt-1).
Next, suppose that we write Z = XY. Then, by noting Δln(Z) = ΔZ/Z, we can easily
realize
Δln(XY) = ΔX/X + ΔY/Y
since Δ(XY) = YΔX + XΔY for small changes ΔX and ΔY. This last equation shows that
the (undifferenced) left and right sides of the equation differ only by a constant. This
means
ln(XY) = ln(X) + ln(Y) + Co.
But, if X = 1 and Y = 1, the constant is seen to be Co = 0. This gives us the important
relation
ln(XY) = ln(X) + ln(Y)
.
To summarize this result, we can say that the ln(X) function turns products into sums
which is a very important property both in mathematics and statistics.
One might well wonder about ln(X/Y). This is easy once we understand multiplication.
Let Z = X/Y. Therefore, X = ZY. Now, take the log of both sides to get ln(X) = ln(ZY).
But, we know this is just ln(X) = ln(Z) + ln(Y), and therefore, since Z = X/Y,
ln(X/Y) = ln(X) – ln(Y)
.
Similarly, the growth rate of a ratio is equal to the difference in growth rates of the
numerator and denominator, or
Δln(X/Y) = ΔX/X – ΔY/Y.
Yet another interesting and useful property of the ln(X) function concerns the logs of
exponentiated variables, like ln(Xα). This is difficult to relate to the area in the graphs
above. Instead, we assume the reader knows that Δ(Xα) ≈ αXα-1ΔX. Consider now the
following:
Δln(Y) = ΔY/Y
where Y = Xα. This means that
Δln(Xα) = αXα-1ΔX/Xα = αΔX/X = αΔln(X)
Again, we note that both (undifferenced) sides must differ by only a constant, which can
be shown to be zero by assuming X = 1. Leaving us with
ln(Xα) = αln(X)
which is a fundamental and highly useful property of the ln(X) function.
Finally, we should point out that
ln(1+X) ≈ X
for |X| small which is easy to see from the graph below.
Summary of Results: ln(X) defined only for X > 0
(1) ln(XY) = ln(X) + ln(Y)
(2) ln(X/Y) = ln(X) – ln(Y)
(3) ln(Xα) = αln(X)
(4) ln(1+X) ≈ X for |X| small
(5) ln(1) = 0 and ln(e) = 1
(6) Δln(X) = ΔX/X
Note: ln(ex) = x by (3) and (5)
The graph of the ln(X) function looks like the following:
One important type of graph which economists use involves the ln(X) function and
growth rates. Suppose that X is a function of time. We can write this as X(t). If we take
the log and difference this, we get the growth rate of X. This can be graphed as below.
Note that there are 3 types of lines given.
(1) shows a straight line, which means that X has a constant positive growth rate over
time, (2) shows that X once again has a constant growth rate, but the value of X jumps at
one point. This is what is called a change in level without a change in growth. For
example, it is often important to make a distinction between a jump in prices and a
change in inflation. (3) shows a sudden increase in the growth rate of X without a jump in
the level of X. This is often important in discussions of monetary policy and changes in
the growth rate of money. Note how that the increased growth rate is depicted as an
increase in the slope of ln(X(t)).
There are other combinations of jumps and changes in slope that can be made. In the
graph below we assume constant growth rates, except in (3) where the growth rate
suddenly increases. A downward sloping straight line would indicate a constant negative
growth rate. Remember that a variable that declines 10% per year never reaches zero.
Such a variable gets closer and closer to zero. A flat horizontal line would correspond to
a zero growth rate. It is important to understand what is happening when we discuss
growth rates.
Another important application of the log function comes in discussing the equations of
exchange MV = PY. Taking logs and differencing we get
ΔM/M + ΔV/V = ΔP/P + ΔY/Y
The first term is the growth of nominal money, followed by the growth of the velocity of
money. One the right we have the rate of inflation followed by the rate of real economic
growth. Certainly this is one of the more important equations in macroeconomics.