Section 8.6 Properties of Logarithms; Solving Exponential Equations
833
Version: Fall 2007
8.6 Properties of Logarithms;
Solving Exponential Equations
Logarithms were actually discovered and used in ancient times by both Indian and
Islamic mathematicians. They were not used widely, though, until the 1600’s, when
logarithms simpliﬁed the large amounts of hand computations needed in the scientiﬁc
explorations of the times. In particular, after the invention of the telescope, calcula-
tions involving astronomical data became very important, and logarithms became an
essential mathematical tool. Indeed, until the invention of the computer and electronic
calculator in recent times, hand calculations using logarithms were a staple of every
science student’s curriculum.
The usefulness of logarithms in calculations is based on the following three important
properties, known generally as the
properties of logarithms
.
Properties of Logarithms
a) log
b
(
MN
) = log
b
(
M
) + log
b
(
N
)
b) log
b
±
M
N
²
= log
b
(
M
)
−
log
b
(
N
)
c) log
b
(
M
r
) =
r
log
b
(
M
)
provided that
M,N,b >
0.
The ﬁrst property says that the “log of a product is the sum of the logs.” The second
says that the “log of a quotient is the diﬀerence of the logs.” And the third property is
sometimes referred to as the “power rule”. Loosely speaking, when taking the log of a
power, you can just move the exponent out in front of the log.
We won’t go into the details of the computation procedures using properties (a) and
(b), since these procedures are no longer necessary after the invention of the calculator.
But the idea is that a time-consuming product of two numbers, for example two 10-digit
numbers, can be transformed by property (a) into a much simpler addition problem.
Similarly, a large and diﬃcult quotient can be transformed by property (b) into a much
simpler subtraction problem. Properties (a) and (b) are also the basis for the slide rule,
a mechanical computation device that preceded the electronic calculator (very fast and
useful, but only accurate to about three digits).
Property (c), on the other hand, is still useful for diﬃcult computations. If you try
to compute a large power, say 2
100
, on a calculator or computer, you’ll get an error
message. That’s because all calculators and computers can only handle numbers and
exponents within a certain range. So to compute a large power, it’s necessary to use
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
1