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

2.4.4 Logarithmfunctions
2.4 Review of some common engineering functions and techniques 85
Logarithms
The equation 16 = 2 4 may be expressed in an alternative form using logarithms. In
logarithmic form wewrite
log 2
16 = 4
and say ‘log to the base 2 of 16 equals 4’. Hence logarithms are nothing other than
powers. The logarithmic form isillustratedby more examples:
125 = 5 3 so log 5
125 = 3
64=8 2 solog 8
64=2
16=4 2 solog 4
16=2
1000 = 10 3 so log 10
1000 = 3
Ingeneral,
ifc =a b , thenb =log a
c
In practice, most logarithms use base 10 or base e. Logarithms using base e are called
naturallogarithms.Log 10
xandlog e
xareusuallyabbreviatedtologxandlnx,respectively.
Most scientific calculators have both logs to base 10 and logs to base e as preprogrammed
functions, usually denoted as log and ln, respectively. Some calculations
in communications engineering use base 2. Your calculator will probably not calculate
base 2 logarithms directly. We shall see how toovercome thisshortly.
Focusing on base 10 wesee that
ify=10 x then x=logy
Equivalently,
ifx=logy then y=10 x
Usingbase e we see that
ify=e x then x=lny
Equivalently,
ifx=lny then y=e x
Example2.12 Solve the equations
(a) 16 = 10 x (b) 30 = e x (c) logx = 1.5 (d) lnx = 0.75
Solution (a)
16 = 10 x
log16=x
x = 1.204
(c) logx = 1.5
x = 10 1.5
= 31.623
(b)
30 = e x
ln30=x
x = 3.401
(d) lnx = 0.75
x = e 0.75
= 2.117
Example2.13 Solve the equations
(a) 50 = 9(10 2x ) (b) 3e −(2x+1) = 10
(c) log(x 2 −1) =2 (d) 3ln(4x +7) =12