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

98 Chapter 2 Engineering functions
(c) ln(3y 6 )−2ln3+lny
(d) ln(6x +4) −ln(3x +2)
(e) log(9x) ( 2
−log
2 3x)
4 Sketch graphs ofthe followingfunctions,usingthe
same axes:
y=ln(2x),
y=lnx 0<x10
Measurethe vertical distancebetweenthe graphs for
x = 1,x = 2 andx = 8. Canyou explain your
findings usingthe laws oflogarithms?
5 Solve the following equations:
(a) e x = 70 (b) e x = 1 3
(c) e −x = 1 (d) 3e x = 50
(e) e 3x = 50 (f) e 2x+3 = 300
(g) e −x+1 = 0.75 (h) 2ee 2x = 50
3
(i)
e x +1 = 0.6 (j) 3
= 0.6
ex+1 (k) (e x ) 3 = 200 (l) √ e 2x =2
(m) √ e 2x e x
+4=6 (n)
e x +2 = 0.7
(o) e 2x = 7e x (p) 2e −x = 9
(q) (e x +3) 2 =25 (r) (3e −x −6) 3 =8
(s) e 2x −3e x +2=0 (t) 2e 2x −7e x +3=0
(u) e x (5−e x )=6 (v) e x −7+ 12
e x = 0
6 Solve the following equations:
(a) 10 x = 30 (b) 10 x = 0.25
(c) 4(10 x ) = 20 (d) 10 2x = 90
(e) 10 3x−2 = 20 (f) 3(10 x+3 ) = 36
(g) 10 −3x = 0.02 (h) 7(10 −2x ) = 1.4
(i) 10 x−2 = 20 (j) 10 3x+1 = 75
4
(k)
10 x = 6 (l) (10−x ) 2 = 40
(m) √ 10 4x 10 −x
= 3 (n)
2 +10 −x = 1 2
(o) √ 10 2x +6 = 5 (p) 10 6x = 30(10 3x )
(q) (10 −x +2) 2 = 6 (r) 6(10 −3x ) = 10
(s) 10 2x −7(10 x ) +10 = 0
(t) 10 4x −8(10 2x ) +16 = 0
(u) 10 x −5 +6(10 −x ) = 0
(v) 4(10 2x ) −8(10 x ) +3 = 0
7 Solve
(a) logx = 1.6 (b) log2x = 1.6
(c) log(2 +x) = 1.6 (d) 2log(x 2 ) = 2.4
(e) log(2x −3) = 0.7
8 Solve
(a) lnx=2.4
(b) ln3x=4
(c) 2ln(2x
(
−1)
)
= 5 (d) ln(2x 2 ) = 4.5
x +1
(e) ln = 0.9
3
9 Solve
(a) e 3x = 21 (b) 10 −2x = 6.7
1
(c)
e −x +2 = 0.3 (d) 2e(x/2) −1 = 0
(e) 3(10 (−4x+6) ) = 17
(f) (e x−1 ) 3 +e 3x = 500
(g) √ 10 2x +100 = 3(10 x )
10 Calculate the voltagegain in decibelsofthe following
amplifiers:
(a) input signal = 0.1V, outputsignal =1V;
(b) input signal = 1 mV, outputsignal = 10V;
(c) input signal = 5 mV, outputsignal = 8 V;
(d) input signal = 60mV, outputsignal =2V.
11 An audio amplifierconsistsoftwo stages:a
preamplifierandamainamplifier. Given the
following data,calculatethe voltagegain in decibels
ofthe individual stagesandthe overall gain in
decibelsofthe audio amplifier:
preamplifier:
main amplifier:
input signal = 10mV,
outputsignal = 200 mV
input signal = 400 mV,
outputsignal = 3 V
12 ABluetooth radio system operating in Class3has a
maximumoutputpower of0dBm at2.45 GHz. Find
the maximumoutputpower in watts (W).
13 Amicrowave oven has an outputpower of1kW.
Expressthisfigurein dBm.
14 Express0dBSPLasasound pressure in
micropascals (µPa).
15 Acar audio speaker has an outputsound pressure
level of55dBSPLwhenmeasuredat adistanceof
1 m. Calculate the sound pressure atthatpoint in
pascal r.m.s.
16 An active sonarsystemfitted to aboatproducesa
sourcelevel of220 dBre 1 µPa at1m. Calculate the
sound pressure in kPa.