Deciding Whether Statements about Logarithms Are TrueDecide whether each statement is trueor false.
(a)
Evaluate each side.
Left side Right side
Write 8 and 4 Write 4 as a
as powers of 2. power of 2.Subtract.The statement is false because
(b)
Evaluate each side.
Left side Right sideSimplify.The statement is true because 1 = 1. NOW TRY
= 1 3 = 31 = 1
= loga ax=x
2
2
= log 3 3 loga ax=x
=
log 7 72
log 8 82
= log 3 1 log 2 232
log 7 49
log 8 64
log 3 1 log 2 82
log 3 1 log 2 82 =
log 7 49
log 8 64
1 Z 2.
= 1
= 3 - 2 loga ax=x = 2 loga ax=x
= log 2 23 - log 2 22 =log 2 22
log 2 8 - log 2 4 log 2 4
log 2 8 - log 2 4 = log 2 4
EXAMPLE 7
SECTION 10.4 Properties of Logarithms 601
Long before the days of calculators and computers, the search for making calcula-
tions easier was an ongoing process. Machines built by Charles Babbage and
Blaise Pascal, a system of “rods” used by John Napier, and slide rules were the
forerunners of today’s electronic marvels. The invention of logarithms by John
Napier in the sixteenth century was a great breakthrough in the search for easier
methods of calculation.
Since logarithms are exponents, their properties allowed users of tables of
common logarithms to multiply by adding, divide by subtracting, raise to powers by
multiplying, and take roots by dividing. Although logarithms are no longer used for
computations, they play an important part in higher mathematics.
For Discussion or Writing
1. To multiply 458.3 by 294.6 using logarithms, we add and
and then find 10 to this power. Perform this multiplication using the key*
and the key on your calculator. Check your answer by multiplying directly
with your calculator.
2. Try division, raising to a power, and taking a root by this method.
10 xlog xlog 10 458.3 log 10 294.6,
CONNECTIONS
*In this text, the notation log xis used to mean log 10 x.This is also the meaning of the log key on calculators.Napier’s Rods
Source: IBM Corporate Archives.
NOW TRY
EXERCISE 7
Decide whether each statement
is trueor false.
(a)
(b) 1 log 2 421 log 3 92 =log 6 36
log 2 16 +log 2 16 =log 2 32NOW TRY ANSWERS
- (a)false (b)false
Write 49 and 64
using exponents.Write 8 as a
power of 2.