082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 87
(b) 3logx +logx 2 = logx 3 +logx 2 = logx
( ( ( 5
)
1 1 x
(c) 5lnx+ln =lnx
x)
5 5
+ln = ln =lnx
x)
4
x
(d) logxy+logx−2logy=logxy+logx−logy 2
( ) ( ) xyx x
2
=log = log
y 2 y
( ) 4
(e) ln(2x 3 ) −ln + ln27 ( ) 2x
3
=ln +ln27 1/3
x 2 3 4/x 2
( 2x 3 x 2 )
=ln +ln3
( x
5
=ln
2
4
)
+ln3=ln
( ) 3x
5
2
Example2.15 Findlog 2
14.
Solution Usingthe formulaforchangeofbasewehave
log 2
14 = log 10 14
log 10
2 = 1.146
0.301 = 3.807
Engineeringapplication2.10
Signalratiosanddecibels
Theratiobetweentwosignallevelsisoftenofinteresttoengineers.Forexample,the
outputandinputsignalsofanelectronicsystemcanbecomparedtoseeifthesystem
has increasedthe level ofasignal. Acommon case is anamplifier, where the output
signal is usually much larger than the input signal. This signal ratio, known as the
power gain, isoftenexpressed in decibels(dB)given by
( )
Po
power gain (dB) = 10log
P i
whereP o
isthepoweroftheoutputsignalandP i
isthepoweroftheinputsignal.The
term gain is used because ifP o
> P i
, then the logarithm function is positive, correspondingtoanincreaseinpower.IfP
o
<P i
thenthegainisnegative,corresponding
toadecrease inpower. Inthis situationthe signal issaidtobeattenuated.
The advantage of using decibels as a measure of gain is that if several electronic
systems are connected together then it is possible to obtain the overall system gain
in decibels by adding together the individual system gains. We will show this for
threesystems connected together, butthe development iseasilygeneralized tomore
systems. Let the power input to the first system be P i1
, and the power output from
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