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1018 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
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Note that the logarithm was taken to the base 10—the number to be
taken to the power of x. There is no limitation to the numerical value of
the base except that tables and calculators are designed to handle either
a base of 10 (common logarithm, LOG ) or base e � 2.7183 (natural
logarithm, IN
). In review, therefore,
(23.1)
The base to be employed is a function of the area of application. If a
conversion from one base to the other is required, the following equation
can be applied:
(23.2)
The content of this chapter is such that we will concentrate solely on
the common logarithm. However, a number of the conclusions are also
applicable to natural logarithms.
Some Areas of Application
The following is a short list of the most common applications of the
logarithmic function:
1. This chapter will demonstrate that the use of logarithms permits
plotting the response of a system for a range of values that may
otherwise be impossible or unwieldy with a linear scale.
2. Levels of power, voltage, and the like, can be compared without
dealing with very large or very small numbers that often cloud
the true impact of the difference in magnitudes.
3. A number of systems respond to outside stimuli in a nonlinear
logarithmic manner. The result is a mathematical model that permits
a direct calculation of the response of the system to a particular
input signal.
4. The response of a cascaded or compound system can be rapidly
determined using logarithms if the gain of each stage is known on
a logarithmic basis. This characteristic will be demonstrated in an
example to follow.
Graphs
If N � (b) x , then x � log b N.
log e x � 2.3 log 10 x
Graph paper is available in the semilog and log-log varieties. Semilog
paper has only one log scale, with the other a linear scale. Both scales
of log-log paper are log scales. A section of semilog paper appears in
Fig. 23.1. Note the linear (even-spaced-interval) vertical scaling and the
repeating intervals of the log scale at multiples of 10.
The spacing of the log scale is determined by taking the common
log (base 10) of the number. The scaling starts with 1, since log 10 1
� 0. The distance between 1 and 2 is determined by log 10 2 �
0.3010, or approximately 30% of the full distance of a log interval,
as shown on the graph. The distance between 1 and 3 is determined
by log 10 3 � 0.4771, or about 48% of the full width. For future reference,
keep in mind that almost 50% of the width of one log interval
isrepresented by a 3 rather than by the 5 of a linear scale. In
addition, note that the number 5 is about 70% of the full width, and
8 is about 90%. Remembering the percentage of full width of the