Understanding Logarithmic Charts
Exponential and Logarithmic Functions
The relationship between 16 and 4 can be expressed as
16  42
(1)
2  log4 16.
(2)
or equivalently as
More generally, the relationship between Y and y, described in terms of the following
exponential function,
Y  a y , a  0
(3)
can also be expressed in terms of the following logarithmic function
y  log a Y , a  0.
(4)
Equation (3) is read “a (raised) to the exponent (power) y equals Y,” whereas equation (4) is
read “The logarithm of Y to the base a equals y”. Figure 1 shows two examples of the
correspondence between Y and y.
The following general rules apply to the logarithmic function:
log a XY  log a X  log a Y
(5)
(6)
log a X b  b  log a X
For example, if a  10, b  3, X  10 and Y  100,
log10 (10  100)  log10 1,000  3  log10 10  log10 100


 


(7)
log10 (103 )  log10 1, 000  3  3  log10 10.



(8)
1
2
and
1
1
Figure 1 Correspondence between Y and y
(i) a = 2
1
2
4
8
0
1
2
3
1
10
100
1,000
0
1
2
3
Y
y
(ii) a = 10
Y
y
Interpreting Logarithmic-Scale Charts
Suppose that Yt represents Japan’s GDP per capita in year t. If the country’s GDP per capita is
growing at the constant rate of g each year,1 the relationship between Yt and Yt-1 is
Yt  (1  g )  Yt 1.
(9)
and the relationship between Yt-1 and Yt-2 is
Yt 1  (1  g )Yt 2 .
(10)
Now let us replace Yt-1 in (9) with the right-hand side of (10). Repeating the same
operation for earlier years, we find
Yt  1  g   Yt  2  1  g   Yt  3  ...
2
3
(11)
and hence
1
If the annual growth rate is 5 percent, g = 0.05.
2
Figure 2. Graphs of Yt and yt
Panel(a)
Yt
Y0
t
Panel(b)
yt
y0
t
Yt  1  g   Y0 .
t
(12)
Next take the logarithm of both sides of equation (12) and apply the rules of equations
(5) and (6). Doing so yields
log a Yt  log a  (1  g )t  Y0   t  log a 1  g   log a Y0 .
(13)
Finally, let us rewrite Y0 and Yt as in equation (4):
yt  t  log a 1  g   y0 .
(14)
If both loga(1+g) and y0 are constants, equation (14) is a linear function of t. If we draw
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this equation in a two-dimensional graph, taking t on the horizontal axis and yt on the vertical
axis, it becomes a straight line passing through point (0, y0) with the slope being loga(1+g).
Figure 2 compare graphs for Yt and yt.
In reality, the growth rates of economic variables are not constant but fluctuate over time.
Diagrams like Panel (b) are often useful for distinguishing the long-term trend of an economic
variable from its short-run fluctuations and for detecting a break in its long-term trend.
As an example, Figure 3 plots PPP GDP per capita of the United States, Germany, Japan
and South Korea over the past 150 years. Although the upper panel is useful for comparing the
levels of the four countries’ GDP per capita, the lower panel is more convenient for comparing
their growth rates. According to the lower panel, the growth rate of the USA’s GDP per capita
has been remarkably stable over the past 150 years. Korea’s growth rate, while extremely fast,
seems to have slowed down recently, a change difficult to detect in the upper panel.
We can take advantage of logarithmic scales in a number of other ways. As another
example, Figure 4 plots the amount of (physical) capital and GDP per worker for a large number
of countries. Panel (a), which uses the standard linear scale for both of the vertical and
horizontal axes, is congested near the origin and difficult to distinguish individual countries.
On the other hand, Panel (b), which employs the base-10 logarithmic scale for both axes, is
much easier to interpret. According to this panel, countries with more capital per worker tend
to have more value added per worker, suggesting that building up physical capital is a key to
generating high income.
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Figure 3. PPP GDP per capita in the USA, Germany, Japan and Korea
150
Germany
Japan
125
Korea
USA
100
75
50
25
0
1860
1,000
1880
1900
1920
1940
1960
1980
1920
1940
1960
1980
2000
Germany
Japan
Korea
USA
100
10
1
1860
1880
1900
2000
(Source) Maddison Project Database (http://www.ggdc.net/maddison/maddison-project/home.htm).
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Figure 4. GDP and physical capital per worker (2009)
Panel (b)
Panel (a)
1,000
500
Quarer
Captal per woker (2005 dollars)
400
300
Luxembourg
Luxembourg
Captal per woker (2005 dollars)
Quatar
Brunei
200
Suriname
100
Libya
10
Cuba
Malawi
100
Nigeria
Central African Republic
0
1
0
100
200
300
400
GDP per worker (2005 dollars)
500
1
10
100
1,000
GDP per worker (2005 dollars)
(Note) All values are measured in terms of 2005 PPP international dollars.
(Source) David N. Weil, Economic Growth (3rd ed.), Pearson, pp. 69.
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