Memorial University of Newfoundland
Department of Physics and Physical Oceanography
Logarithmic Graphs
Functions of the kind y = axn occur frequently in physics. The nature of the function
depends on the value of n, as shown in Figure (1)
y
3
x
2
x
2
/
3
3
/
2
x
x
1
/
2
x
1
/
3
x
x
Figure 1: Graphs of the function y = xn for various values of n (n > 1)
The value of n, if not already identified by theory, can be determined by a computer-aided
power law fit, or using logarithmic scales.
Logarithmically scaled graphs allow data to be plotted without having to calculate the
logarithm at each data point. The axes of the graph are scaled according to the logarithm
(base 10) of the number rather than the number itself. There is no origin because log 0 is
not defined. Figure (2) shows plots of y = xn on a log-log scale for various values of n. In
each case a straight line is obtained of slope
n=
log10 y2 − log10 y1
.
log10 x2 − log10 x1
Determining the slope is simplified when y2 and y1 are chosen to be a whole number of
cycles apart. So if y2 = 100 and y1 = 10, log10 y2 − log10 y1 = 2 − 1 = 1. Similarly, when
the logarithm is negative, e.g., y1 = 10−5 and y2 = 10−3 , the difference in the logarithms is
(−3 − (−5)) = 2. On log-log paper the cycles on both axes are usually square and the slope
is simply equal to
distance along y-axis (in mm, cm, in, etc)
,
distance along x-axis (in mm, cm, in, etc)
regardless of the values of x and y.
1
10000
y=x 3/2
y=x 3
y values
1000
y=Ax
100
-1/2
10
1
1
10
100
x values
1000
10000
Figure 2: log-log plots of y = xn for various values of n
Fig (3) shows the voltage measured across a light bulb as the current is increased. V is
proportional to I n , where the value of n is obtained from the ratio
6.5 cm
∆y
=
= 1.8.
∆x
3.6 cm
[These lengths may not be exact due to some distortion in the printing process.] Hence
V = A × I 1.8 where A is a constant, determined from the y intercept. On a log-log plot
the intercept is the y value at x = 1 (i.e. log x = 0). It can be seen from Fig (4) that the
intercept is 0.003, and hence V = 0.003I 1.8 .
Semilogarithmic Plots
Functions of the kind y = Beφx are used to describe the absorbtion of electromagnetic
radiation or radioactive decay, and may be plotted on a semilogarithmic scale. Taking log10
of both sides gives
log10 y = log10 B + φx log10 e = log10 B + 0.434φx.
Plotting log y vs x will therefore give a straight line of slope 0.434 × φ. [Question: Why
does the function y = Beφx + c not give a straight line on a semilogarithmic scale?]
2
100
10
∆ y = 6.5
Voltage across bulb (V)
10
1
1
∆ log10V=3
Voltage across bulb (V)
100
0.1
∆ x = 3.6 cm
0.01
∆ log10I=1.65
0.004
0.003
0.1
1
0.002
10
100
Current through bulb (mA)
0.001
Figure 3: The slope can be obtained from the ratio of the lengths
∆y/∆x = 1.8
Intercept=0.003
1
10
100
Current through bulb (mA)
Figure 4: The y intercept is the y
value at x = 1. Here, the intercept
is 0.003 and hence V = 0.003I 1.8 .
3
1000
The slope is determined in the usual way,
slope =
log y2 − log y1
x2 − x1
with the y-values chosen such that the difference in their logarithms (base 10) is a whole
number.
Fig 5 shows data which was obtained study the the decay of Ba-137m. The number of
atoms remaining at time t is described by the expression N (t) = N◦ e−λt , where λ is the
decay constant. The slope of the graph is (−)1/507, and hence
λ=
1
÷ 0.434 = 0.0045 s−1 .
507
The half-life is given by 0.693/λ = 152 s.
10000
1000
counts
∆ log10y=1
100
∆ x=507
10
0.0
200.0
400.0
600.0
Time (seconds)
800.0
Figure 5: Data to show decay of Ba-137m
Note: Not all graphs plotted on logarithmic scales are straight lines. An obvious
example is the frequency response of a filter from frequencies as low as a few Hz up to several
tens of kHz. This kind of data is usually presented on a logarithmic scale because it would
be impractical to draw such a plot on a linear scale.
4