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Mathematical Methods Unit 2
Worksheet 1 — Negative & Rational Powers
1 (QV Express each of the following with positive index numbers.
2 (ETM IFW Simplify cach of the following, expressing your answer with positive index numbers.
(" 2 x 24 (x? Y
P x Qt)? Cm) x? m
F
23 re xO AY : (p?)!xp 4
7 x?. (x 4 ) 4 ( 3 - 7 )? x( 2 ° 5 )! wy? x (xy?) 3
x 3 () 3 * Oy <iay* ( 2 x 3 )? x(y 3 )?
3 f& Evaluate the following without a calculator.
1 1 2 3
a 92 b 273 c 83 d 814
3 4 é 2
16 \ 4 25 ) 2 4 £.) 3
« (F)
+ 8 9 8! n (a)
4 rel 25 x 125 simplifies to:
5 7 3 u 8
A 256 B 56 Cc 52 D 56 — 56
5 Simplify each of the following, expressing your answer with positive indices.
a VOx BI b x 3 xx 6 c Y(xy+ ¥?(x)?y)
5 1 2 6 ) 3 I
d 24 x 42 x 83 e (o 4 mnlys ' +z
4 m? x
(x + 1 )? =
h (-~ 4 )yy- 4
9 vxt 1 ’ v
up-Q 4 +x) 6
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Mathematical M
ethods
Unit 2
Worksh
eet 2 — Solvin
g Indicial Equa
tions
Short answer
@ 1 Solve for x.
(a) 2 *= 64 (b) 3 *= 27 (c) 8 °= 2
(d) 64 *= 2 (e) 3 *= 81 (f) 2 % = 4
(g) S* = (h) 3 % = 5 h @) 5 % = 25
@ 2 Solve for x.
(a) Qax+l= 16 (b) 53 x- 1 = 125 (c) 5 l-ax = *
(d) 32 x- 2 = x (e) 64 x 43 = 162 - 1 (£) 25 3 x 41 = 6252 - 3
(g) 5 *x 2541 = 125 % (h) 9 x 27 *+ 3 = ( 31 ) 4 QQ) 16 -= 2 %. 32 t- 1
& 3 Solve for x.
(a) 5 %*- 4 ( 5 )- 5 = 0 (b) 27 ° 12 ( 2 *) + 32 = 0 (c) 2 ( 9 ) - 5 ()- 3 = 0
(d) 29 ") - 4 ( 3 *)- 6 = 0 (e) 2 ( 2 %) - 5 ( 2 )+ 2 = 0 (f) 2 ( 2 %) - 8 ( 2 ) + 8 = 0
(g) 5 ( 5 %) — 1066 ") = - 5 (h) 3 %) — 18 ( 3 ) = - 27 @ (G- 9 )( 5 *- 1 )= 0
Multislie choice
4 If 3 *+ 2 x 27 % = 9 % thenx 1 s equal to:
A 0 BS cl D - 3 E - 2
5 Given 6 ( 3 ”) — 15 ( 3 *) = 9 , then x is equal to:
A 0 Bil Cc - 1 D 3 , - 3 E 3
Extended answer
6 Two different substances decompose at
different rates. For substance A, the
quantity left after a period of time t is 2
and for substance B it is 4 !**,
(a) By substituting t = 0 , find the
amount of each substance there is
to start with.
(b) When would there be the’same
amount of each substance?
(c) If both substances are present in
our garbage, which one is better for
the environment, assuming no
toxic substances are produced?
7 if
b=-a? + 6
a— 8 then:
@ (a) facto
rise the expressio
n and d
etermine the value
s o
f a when b = 0
(b) if a = 2 %, re
write the express
ion in terms of b and
x
(c
) for what va
lues of x does
b = 0?
L=X 7 =x Q)
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Mathematical Methods Unit 2
17 6 ) The expression logs (x”) is equal to:
A x logs (y) By logs (x) C 5 log, (y) D logs(x)+logs(y) & »
12. (i{@] The expression z logy ( 64 ) + log, ( 5 ) can be simplified to:
A. log ( 40 ) BI C log ( 2 ) D log, ( 20 ) & loge ( 2 °)
_ logy (x°
13 (Hi) The expression 1084 0 ") can be simplified to:
logy (x?)
5
A logy (x?) B logy (x?) c 3 D logy (x? — x?) E logy (x)
14 Express each of the following in simplest form.
a log; ( 27 ) + 1 b log, ( 16 ) + 3 c 3 logs ( 2 )- 2
d 243 logig (x) @ 4 logs ( 2 )~ 2 logs ( 6 ) + 2 Ff $ 43 logig(x?)
15 [MEE Evaluate the following, correct to 3 decimal places where appropriate.
a log, ( 16 ) b log, ( 81 ) ¢ logy ( 0. 001 )
d logs ( 9 ) e log; ( 4 ) + log ( 7 )
oor ® Le p
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Mathematical Methods
Unit 2
Worksheet 4
~ Solving Logarith
mic Equations
- Solve for x (to three decimal places).
(a) log, x = 3. 1 (b) log, x= 2. 5 (c) logs («+ 1 ) = 2. 7
(d) log, @ + 2 ) = 2. 4 (e) 3 log; 2 x= 1 (f) logs 2 x = 3. 4
2 Solve for x.
(a) logy 1000 = x (b) logy 1 =x (c) log, 27 =x
(d) log, 16 =x (e) log, 625 =x+ 1 (f) log, 16 = 4
(g) 2 log, 9 = 2 x (h) log, 243 = 2 x- 1 (i) log, 8 = 3. 5
Gj) 3 log, 4 = 3 x+ 1 (k) 3 log, 4 = 6 () 2 log, 3 = 4
3 Solve for x.
(a) logy («+ 1 ) — logig 2 = 1
(c) logs 2 x + logs 6 = logs ( 4 x + 8 )
(e) logig 2 x? = logy, 50
(g) logyo (x? +X) — logig ( 8 x — 1 )
(i) 2 logio x —logig x= 1
(k) (logig x)* — 2 logigx- 3 = 0
(an) 2 (logio x)? — 7 logy x= 4
multiple choice
4 Given that log, x + logy (x + 4 ) -
AS B - 5
5 If (logio x)? — 3 logig x + 2 = 0 then x is equal to:
A 2 , 1 B 10 , 100
extended answer
6 The concentration of a solution,
C, in M (moles per litre), at time
f minutes, in a sample as it is
dissolving is modelled by the
equation C = log, 0. 35 (t + 3. 4 ).
(a) Find the concentration at
the start, i.e. f= 0 , in exact
form.
When the concentration of
the solution is equal to
2 , it is dissolved. Find
the time that this occurs.
Another sample has a
concentration of
C = log, 0. 3 (t - 4 ). At what
time, if at all, will the
concentrations of the two
samples be the same?
(b)
(c)
logy ( 4 x + 25 ) =
Cc
Cc 1 , 10
(b) logig ( 2 x + 1 ) — logy 3 = 2
(d)_ logs x + logs (x + 5 ) = log, ( 5 x + 9 )
(f) logs 3 x? = logs 27
(h) logyo ( 2 x? + 3 x + 3 ) — logyy ( 1 — 2 x) = 0
Qj) 2 loxg — slogs - = 1
(I) (logyg x)? — 2 logiyx- 8 = 0
(n) 2 (logyo x)? — login x = 6
0 , x equals: :
p - 24 2.
Li ,
D=,— E 0 , 1
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Mathematical Methods Unit 2
Worksheet 6 - Exponentials & Logarithms with base ¢
Question 1
Simplify each of the following.
3 n 5 - 2 n
3 e”" x 2 e
a e'x
(e*"")
b. ey
3
4 e°a°_[{ &?"a’ 1 ,
a 3 n- 4 d. —+e
3 ea~ 2 ae e
Question 2
Solve for x in each of the following.
a. er! _ ek b 1. = es xe 7 x
om
1 _ sy 2 - 3 em
C. == 4 (e 8 *) d. (e ) - —
z oe
Question 3
Solve for x in each of the following, correct to three decimal places.
a e= 2 b. ihe
Question 4
Simplify the following.
a. log,(e)' b. log, ( 25 ) +log, ( 125 ) - 2 log, ( 25 )
2 log, ( 36 ) g 3108. ( 64 x)
- log, ( 216 ) "Jog, ( 8 x)
Question 5
Solve for x in each of the following.
a. log, (x)= 5 b. et a 4
c. 2 log,( 2 x- 1 )= 6 d. log, ( 6 - 2 x)= log, ( 3 x+ 2 )
e. ede" =- 3 f. 2 log,( 2 x)-log, (x + 1 ) = log, ( 4 )
2 x
g. 363! = 5 e" h. <_= 6
é€
i. log,( 2 )= 4 x- 1 j. alog,( 3 x+ 2 )= 5 , where a Is positive number
Worked solutions on the google drive.
Mathematical Methods Unit 2
Worksheet 7 —- Graphs of Exponential and Logarithmic Functions
Question 1
Describe the transformations from the basic function y = /'(x) to the transformed function
y=g(x).
a. f(x)=e" g(x)=- 2 e"- 1
c. f(x)=log.(x) g(x)= 3 log,( 2 x+ 1 )
d. f(x) = 2 - log,(x+ 4 )
Mathematical Methods
Unit 2
c. f(x)= 2 - 3 e"
y z
t if
2 2
pp tt
{ om |
— [>
4 2 2 4 6 8 10
4 2 4
10
2 - 24
44 4 +
6 6
14 10.
Vv
\
le. g [- 1 , 3 ) > R,
g(x) = 3 - e""
f. h:(> 0 R,A,(x) 00 =-log, )
(x) + 2
y y
el
cr —— pds
z 1
1 2 3 4 5
|
6
|
4
10 |
|
i
a
—
1 \ 1 2 5
20 |
ay
Mathematical Methods
Unit 2
g. k:R>Rk(x)=e* 3
y
.
wo) +]
a ;
a 4 :
‘= t * o — 1 a)
: 5 2 a 4 : 3 4
eae a a "|
a|
4!
v s|
i, #:( 3. 8 ] > &. 9 ( 2 )= 2108 , 2 - 5 )
j h:[- 4 , 3 ) > R,h(x)= 1 - e"?
k
Hl
x
5
i “|
1 femora f= t t t » 1 4 t t t + fe
i $ 4 3 ijt oto 2 a aos
a 3 |
at 44
ef i
Mathematical Methods Unit 2
Worksheet 9 - Applications of Exponential & Logarithmic Functions
1 QE Before a mice plague that lasts 6 months, the population of mice in a country region is
estimated to be 10000. The mice population doubles every month during the plague. If P represents the
mice population and 7 is the number of months after the plague starts:
@ express P as a function of¢
b find the population after:
i 3 months ii 6 months
¢ calculate how long it takes the population to reach 100000 during the plague.
2 The population of a town, N, is modelled by the function N = 15 000 ( 2 °°"), where ris the
number of years since 1980 ,
a Find the population in 1980.
b Find the population in:i 1985 it 1990.
¢ What is the predicted population in 2015?
ds In what year will the population reach 20 000?
3 The weight ofa baby, W kg, ¢ weeks after birth can be modelled by W= 3 logyg ( 8 / + 10 ).
a Find the initial weight.
b Find the weight after:i 1 week ii 5 weeks ii 10 weeks.
¢ Sketch the graph.
cd When will the baby reach a weight of 7 kg?
4 If $A is the amount an investment of $P grows to after 7 years at 5 % p.a. using compound interest:
a write A as a function of P
b use the function from @ to find the value of $ 10 000 after 10 years
¢ calculate how many years it will be until an investment of $ 10. 000 reaches $ 26 500.
5 The value of a car, $V, decreases according to the function V = 25 000 ( 2 >", where ¢ is the number of
years since the car was purchased.
a Find the value of the car when new,
b Find the value of the car after 6 years.
¢ In how many years will the car be worth $ 10 000?
6 The temperature, T (°C), of a cooling cup of coffee in a room of
temperature 20 °C can be modelled by 7 = 90 ( 3 °°") , where
tis the number of minutes after it is poured.
a Find the initial temperature.
6 Find the temperature:
i 3 minutes after pouring ii 6 minutes after pouring.
c How long is it until the temperature reaches half its initial
value?
7 A number of deer, NV, are introduced to a reserve. The deer population can be predicted by the model
N= 120 ( 1. 1 '), where¢ is the number of years since introduction.
a Find the initial number of deer in the reserve.
b Find the number of deer after:
i 2 years
ii 4 years
iii 6 years.
¢ How long does it take the population to treble?
d Sketch the graph of N versus ¢.
e Explain why the model is not reliable for an indefinite time period.
8 After a recycling program is introduced, the weight of rubbish disposed of by a household each week
is given by W = 80 ( 2 °°''), where W is the weight in kg and ¢ is the number of weeks since recycling
was introduced.
a Find the weight of rubbish disposed of before recycling starts.
b Find the weight of rubbish disposed of after recycling has been
introduced for:
i 10 weeks it 40 weeks.
c How long is it after recycling starts until the weight of rubbish
disposed of is half its initial value?
d i Will the model be realistic in 10 years time? ii Explain.
Mathematical Methods
Unit 2
? T he number of hectares (WV) of forest land dest royed by a fire ¢ hours
afler it started is given by N = 40 logy ( 500 f-++ 1 ).
a Find the amount of land destroyed after:
i 1 hour
fi 2 hours
tii, 10 hours .
© How long does the fire take to burn out 155 hectare s?
10 A discus thrower competes at several compeutions during the year. The best distance, d metres,
that he achieves at each consecutive competition is modelled by d= 50 + logig ( 15 n), where n is the
competition number.
a Find the distance thrown at the:
i lst competition ii 3 rd competition ili 6 th compeution iv [ 0 th compeution.
b Sketch the graph of d versus n.
¢ How many competitions does it take for the thrower to reach a distance of 53 metres?
11 The population, P, of a certain fish ¢ months after being introduced to a reservoir is
P = 400 ( 10 °°S), 0 <¢< 20. After 20 months, fishing is allowed and the population is then modelled by
P= 15000 + 924 logig [ 10 (¢— 19 )], ¢ 2 20.
a Find the initial population.
6 Find the population after:
i S months ii 15 months
iii 25 months — iv 40 months.
¢ How long does it take the population to pass 10 000?
12 A ball is dropped from a height of 5 metres and rebounds to a of
its previous height.
a Find the rule that describes the height of the ball
(A metres) after n bounces.
b Find the height after: i 4 bounces — ti 8 bounces.
¢ Sketch the graph of the height of the ball after 7 bounces.
13 A computer appreciates in value by 10 % per year. If the computer
costs $ 5000 when new, find:
a the rule describing the value, V, of the computer at any time,
t years, after purchase
b the value of the computer after 6 years
c the number of years it takes to reach double its original value,
14 From the start of 1996 , a small mining town has seen a steady increase in population unul 2000 as the
price of minerals improved and mining was extended.
_ Year | _ 1996 | 1997 i
Population (P) | 700 | 750 _ 304
a Let 1996 be ¢= 0 ; then 1997 will be ¢= | and so on. Plot P against ¢.
b What does the shape of the curve look like?
¢ Calculate the ratio of the population in 1997 to the population in 1996 ,
d Calculate all the ratios in successive years, and hence estimate the percentage annual increase or
1999 | 2000 S
growth.
To obtain an accurate estimate of population growth, follow these steps.
e On the table above, evaluate logyy (P).
f Plot logi (P) against ¢. Are the points approximately collinear?
g Draw a line of best fit and find its gradient and the intercept on the y-axis.
h Write the equation for the line.
i Show that P = 700 ( 1. 07 )!. Is this close to your estimate in d?
j Use this formula to estimate the population in 2001 and 2002.
k When might the population have reached 2000?
! In fact there was a downturn in the population as the mine output decreased. From 2000 onwards
there was an annual decline of 10 % in population. During which year did the population reduce to
below 600?
Mathematical Methods
12 a h= 5 ( 0 , 7 ")
1 a P= 10000 ( 2 ')
b i 80000
ii 640. 000
¢ 3. 3
2 mo
nths
2 a 15000
b i
15528
i 16077
c 1911
8
d 2
3 a 3
kg
;
bi 377 kg
wi
S.Lkg
ii 5. 86
kg
© W
bene
, 1
0 )
ee
d 26 weeks
40 A=P( 1. 05 )"
b$ 16288. 95
¢ 20 years
5 a $ 25
000
b$ 14. 427
c 10
years
6
a 9
0 °C
bi
76. 3 °C
— ii 64. 7
°C
c¢ 12
min 37
s
7 a 120
bi
145
ii 176
iii
213
c Uy years
J "
N= 120 ( 1. 1 '
)
1204
Of
i)
e
The popu
lation wi
ll reach a
limit at
s
ome s
tage.
8 a 80
kg
bi 72. 1 kg
¢ 67 wee
ks
diy "t
3 —>;
i 52. 8 kg
ii No, the
model su
ggests vir
tually no
rubbish
will be
disposed
of in 10
years
or so
, which i
s unlikely.
ga
i 108
hectares
ii 12
0 he
ctares
iii 148
hectares
b 15 h
joa 1
- 48 m
Wi 51. 65 m_
fii
- 95 m
iv 52. 18 m
b
52
S 17 ” d= 50 +
log
g( 15 n)
W 234 &
5
c 67
- 4
400
bi 1005
fi 6340
iii
16643
iv 17146
c¢ 17. 48 months
bi 1. 20 m i 0. 29 m
13 a V=S 000 ( 1. 1
b $ 8857. 81
c 8 years
144 pypeople)
900
800
700
0 1 2 3 4 aytars)
b The graph shows a slight upward curve
not starting from ( 0 , 0 ); this suggests
possible exponential growth.
ce 1. 071
d 1. 072 , 1. 082 , 1. 063. Estimated% annual
growth is 7 %.
Rp a b
e I ~ 4
? : 2. 845
¢ logio(?),
1
0
28
The points appear collinear.
g m= ( 0. 03 ; y-intercept = 2. 845
h logig (P) = 0. 03 + 2. 845
P = 110. 03 + 2. 845 )
=> P = 100 - 03 y¢ 102 - 845
= P= 700 x 1. 072 ", i.e. 7. 2 % growth
similar to d
j Ps = 982 people; 2002 (t= 6 ),
P¢= 1051
During 2011
During 2004
15 a $ 000 ),
50
40
=
=
0 1 2 3 4 x(year)
b The curve starts from $ 45 000 , curving
down with a reducing slope —
exponential decay.
c 0. 789 , 0. 789 , 0. 804 , 0. 800
d The ratio is about 0. 795 or a 20. 5 % rate
of depreciation.
er —
1 2 B
- 550 |aaa 7 4. 352
ls |
- 255 |
Unit 2
Flog (V).
- 6
44
o 12 3 4 6
‘The points are roughly collinear.
g m= ~ 0. 099 ; y-intercept = 4. 65
h logy (V) = " 0. 0 + 994 .¢ 65
i logy (V) = “ 0. 0991 logi 9 ( 10 )
= logyg( 10 9 )
=> Vs 100. 099 y¢ 194. 65
= 104 - 6 x ( 1970 - 099 )
j The rate of depreciation is about 21 %.
k In 2010 (x= 5 ), V= $ 13 847 ; in 2011 ,
V=$ 10939.
! During 2012
lea 4 ,
5000 ]
4000 }
2000 }
1000 }-
0 , 0 ) 1 2 3 4 5 R
The graph starts from ( 0 , 0 ) and curves
slightly up to the right like a positive
exponential function.
logig R logig T
| 0410 1943 |
- 140 2. 352
Foo 2. 563
; ass. | 2. 837
- 716
oo
wA- 3 - 2 - 1 0 - 1 - 2345 6 - 7 - 8 LOiQB (R)
The points are very close to collinear.
d m= 1. 50
e The y-intercept is 2. 5614 or log jg ( 364. 25 )
f logio(T)= 1. 50 logio (R) + 2. 5614
g logiy ( 7 ) = logo ( 364. 25 x RE)
= T= 364. 3 RI
h T= 10711. 76 days. The difference may
be due to rounding errors or ignoring the
small effects of other planets and moons
on Saturn’s orbit.
1 k= 133. 407. 5625
