St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Name: _____________________________ Class (No.): __________(_____) Teacher: Mr Luca Lee YL
6.0
Indices revisited
Indices
(i)
Law of integral indices
$ *+,-.
(i)
- = 344454446
- × - × ⋯× $
(ii)
-: base
-# × -$ = -#%$
&!
&"
= -#'$
7: index/ power/ exponent
(iii)
(-# )$ = -#$
(ii)
-/ = 1
(iv)
(-/)$ = -$ /$
(iii)
-'$ = &"
(v)
0( 1 = ( "
0
Revision Exercises:
Try questions on P.6.5 (textbook) if you forgot the law of indices.
6.1
& $
&"
Review Ex Q10 12
P6.5
Law of Radical Indices
Motivation
If 8 1 = -, then 8 = ±√-, i.e., 8 = √- or 8 = −√-.
-
The value of 8 is called the square root of -.
-
Each positive real number - has two square roots.
Naturally, there will be a similar situation for different powers, e.g.
-
8 ! = - ⟹ The value of 8 is called the cube root of -.
-
8 2 = - ⟹ The value of 8 is called the fourth root of -, and so on.
Definition:
"
"
If 8 $ = -, then 8 is called the =34 root of -. The 756 root of - is denoted by √-. The symbol √
"
is called a radical sign, and √- is generally called a radical.
Example 6.0
!
√64 = 4
!
∵ 4 = 64
∴ 4 is the cube root of 64.
∵ 5" = (−5)" = 15 625
∴ 5 and −5 are the sixth root of 15 625.
"
√15625 = 5
−√15625 = −5
"
1
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Example 6.1 (Refer to Example 6.1 from 4B06)
Relation:
Find the values of the following.
#
$
(a) √81
Case 1: 7 is odd and - is any real number:
%
(b) √−216
0
8$ = -
(c) ?!1.
8$ = -
#
(a) 32 = 81 ⟹ 3 = √81
(b)
0 7
8 = √-
Case 2: 7 is even and - ≥ 0 :
Solution:
(−6)!
"
↔
!
= −216 ⟹ −6 = √−216
0
8 = ± √-
(In this case, if 8 < 0, then √8 is not a real
$
0
"
↔
number.)
0
%
(c) 011 = !1 ⟹ 1 = ?!1
Using your calculator:
"
If you want to compute √-, you input:
7
Exercise 6.1a
-
Find the values of the following.
(a)
"
√10 000
(b)
$
$
(c)
√−27
0
?"2.
Exercise 6.1b
Find the values of the following.
(a)
ca
4
256
4ft
(b)
lb
4
3
- 1000
(c)
FA
o
10
3
-
8
27
Fy
n
fly
Exercise 6.1c (Refer to Example 6.1 from 4B06)
Find the values of the following.
(a)
( 4 16 ) 2
5
(b)
- 243
3
27
(c)
3
8 2 ´ 3 125
2
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
6.1.2 Rational Indices
Motivation
When we take the square root, we cancel the square, i.e., √21 = 2. The same hold for the other
$
$
roots, e.g., √−27 = N(−3)! = −3.
Now, look at the example √625 = √251 = 25. In fact, if we look at this example carefully, we
$
$
$
$
see that √625 = √52 = 51 . Similarly, √19683 = √27! = 27 and √19683 = √3< = 3! = 27.
This suggests that by taking the =th root, the index is divided by =.
Definition:
"
&
Let 7 be a positive integer, then √- = -" . Note that this only holds for the following cases:
Case 1: If 7 is odd and - is any real number, and
!
"
As a result: √-# = - "
Case 2: If 7 is even and - ≥ 0.
& '=
Example 6.2 (Refer to Example 6.2 from 4B06)
Think faster: 0 1
(
( =
=0 1
&
Find the values of the following without using a calculator.
(a)
&
'
(b)
64
'
$
125
$
(c)
2 ''
0< 1
Solution:
&
(a)
64' = √64 = 8
(b)
125! = F125! G
0 1
1
Proof:
0 1
! )!
= H(5
I
= 51
= 25
(c)
!
'
1
4
F G
9
!
1
9
=F G
4
0 !
9 1
= LF G M
4
ere
fact h
e
h
t
i ng
Apply
- '=
1
0 1 =
- =
/
0 1
/
1=
=
- =
0/ 1
=
1
=L - M
0/ 1
/ =
=F G
-
3 !
=F G
2
27
=
8
3
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.2a
Find the values of the following without using a calculator.
(a)
&
$
64
(b)
$
#
$
0 ''
0!"1
(c)
8
Exercise 6.2b
Find the values of the following without using a calculator.
(a)
81
1
4
(b)
32
3
5
æ 64 ö
ç
÷
è 125 ø
(c)
-
2
3
Example 6.3 (Refer to Example 6.3 from 4B06)
Assuming that -, / > 0, simplify the following expressions and express your answers with positive
indices.
(a)
$
√& '
√& $
(b)
N- × √$
(c)
$
√-/1
$
#
÷ 027- /1
'
'
$
Solution:
General approach:
(1) Convert all the roots into rational indices
(2) Apply the law of indices
(3) Express your answer with positive indices ß You will see this question in DSE every year.
Steps (2) and (3) are standard steps in solving questions of this kind.
4
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
$
√& '
(a)
√& $
0
$
√-1
√-!
=
(-1 )!
0
(-! )1
1
=
-!
(-# )$ = -#$
!
-1
1 !
-# ÷ -$ = -#'$
= - !'1
7
= - '"
=
1
Convert to positive indices by flipping
*
it upside down, i.e., !() = +!
7
-"
N- × $√-
(b)
?- × √- = F- ×
$
0
0 1
-! G
=
0
0 1
F-0%! G
=
0
2 1
F-! G
-# × -$ = -#%$
(-# )$ = -#$
1
= -!
$
$
#
(d) √-/1 ÷ 027- /1
'
Think faster:
'
$
1
!)
= # × !)
# ÷ !() = # ÷
$
N-/1
÷
1
'!
!
F27-2 /G
=
=
$
N-/1
1
!
!
× F27-2 /G
1
2 3
)-/ *
×
In short, ÷ !() =× !)
2
3
3
+27-4 /,
1 2
2 3 2
= +-3 /3 , × +273 -4×3 /,
=
1 2
+-3 /3 ,
×
1 1 2
Ob
Ob
1
+9-2 /,
= 9-3+2 /3+1
5 5
= 9-6 /3
5
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.3a
Assuming that -, / > 0, simplify the following expressions and express your answers with positive
indices.
(a)
0
√- ⋅ #√&$
(b)
N- × √-
(c)
s
ax at
at 4
at
$
g
at
g
s
'
&
?16-/$ ÷ 0-$ /1
'
$
'
ibab
at
x
4at
b's
b
at b
x
4abt
Exercise 6.3b
Assuming that 8, S > 0, simplify the following expressions and express your answers with positive
indices.
(a)
x3
3
x2
(b)
4
x ´ x
3
(c)
3
1
2
x y ÷ (8 x y )
2
4
3
-
2
3
6
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.3c
3
Assuming that -, / > 0, simplify
1 3 2
æ a2 ö2
2
2 2 3
ç
÷
a ´ç
÷ ÷ (a b ) . Express your answer with positive
b
è
ø
3
indices.
a
p
3
a
It
Example 6.4 (Refer to Example 6.4 from 4B06)
Relation:
Solve the following equations for 8 > 0.
(a)
$
8' = 8
(b)
!
F" = H
$
"
8 '# = 27
F = H!
Proof and idea:
Solution:
We want to get rid of
(a)
!
81
!
"
on the LHS, so we
"
!
!
"
should multiply it by , so that
i.e.,
=8
8=
×
!
F" = H
1
8!
"
! !
"
IF " J = H !
1
8=2
"
F = H!
8=4
(b)
!
8 '2 = 27
2
8 = 27'!
8 = 3'2
8=
1
81
7
"
!
= 1,
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.4a
Solve the following equations for 8 > 0.
(a)
#
8 $ = 16
a
(b)
x
16
x
164
x
$
8 '' =
1K
L
b
3
x
4
x
8
Iq
x
Exercise 6.4b
Solve the following equations for 8 > 0.
2
3
x = 25
(a)
a
x
25
x
ins
(b)
x
-
3
5
=
b
1
8
x
H
321
8
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Selected Exercises:
Further Practice P.6.14
Exercise 6A
Level 1: Q16-26
Level 2: Q33-53
End of Section Exercises
Q1.
Find the values of the following radicals without using a calculator.
(a)
3
(e)
83
27
1
8
(b)
3
(f)
16 4
1
(i)
(m)
Q2.
(c)
(g)
32
(k)
æ 1 ö6
ç ÷
è 64 ø
(o)
( 5 ´ 25 )
1
216
-
2 ´4
1
5
(d)
64
(h)
25 2
(l)
æ 4 ö
ç ÷
è 25 ø
(p)
æ32 ö
ç
÷
ç 3 16 ÷
è
ø
3
1
2
3
2
3
-
(j)
1
6
(n)
4
10 000
1
4
9 ÷3
-
9
2
2
3
4
8
5
-
3
2
-4
Use a calculator to find the values of the following radicals, correct to 3 significant figures.
(a)
(e)
Q3.
16
625
4
6
42
0.7
-
3
5
- 3.5
(b)
7
(f)
æ8ö
ç ÷
è9ø
-
(c)
9
-
1
11
7
(d)
24 4
2
7
Simplify the following expressions and express your answers with positive indices. [***]
(a)
(e)
x ÷x
5
( a -9 )
2
2
3
1
3
(b)
y ´y
(f)
æ 1
ç
çç 1
è a2
ö
÷
÷÷
ø
1
4
2
(c)
( x y)
3
(d)
(g)
a´6 a
(h)
æ 34 ö
ça ÷
ç ÷
è ø
8
-5
4
8
a
a3
3
(i)
3
8a ÷ 43 a 2
(j)
3
a 2 ´ a3
b
(k)
æ 3 a 2 ´ a -2 ö 4
÷
(l) ç
4
ç
÷
b
è
ø
4
1
6 3 3
(a b )
3
(m)
5
a
3
(n)
æ - 16
ö
ç a ´ a 3 ÷ (o)
ç
÷
è
ø
(r)
1
æ
ö
p
2 p q ´ çç p - 2 q 3 ÷÷ ÷
9
è
ø
-2
a -3
1
(3 27t ) 2
÷ 9t 6
(q)
3
t
1
2
3
x ´
5
x
-
3
4
x
1
(p) (mn) 3 ÷
m4
(3 n ) 2
3
-2
9
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Q4. Simplify the following expressions.
( 2) ´8
(a)
x
5
x
6
(b)
3(9 x )
27 x
Q5. Solve the following equations for 8 > 0.
1
3
(a)
x =4
(c)
x 3 = 36
(b)
x
(d)
x
(f)
5
2
2
3
3 x = 48
(e)
-
1
5
-
3
2
=2
=
1
27
( x + 1) 2 - 4 = 0
Past Paper (Paper 1) [Occurrence: High, Section A(1) (almost) every year]
HKCEE 1987(A) Q3(a)
!
Simplify ?
%,-'
1K,
HKCEE 1999 Q1
'
Simplify
.
O& .$ P
and express your answer with
&
HKCEE 1990 Q2(a)
positive indices.
&
HKCEE 2000 Q2
Simplify
√&
, expressing your answer in index
M .$ N
form.
Simplify
HKCEE 1993 Q5(a)
positive indices.
Simplify and express with positive indices
HKCEE 2001 Q1
M .&
M'
and express your answer with
#$
'!
8 0 N' 1 .
Simplify (#$)' and express your answer with
HKCEE 1994 Q7(a)
positive indices.
Simplify
O& # ( .' P
'
&(
and express your answer
with positive indices.
HKCEE 1996 Q2
Simplify
%#
& # √& $
& .'
M$N'
M .$ N
.
HKCEE 1998 Q4
Simplify
( .'
O&( ' P
'
and express your answer with
&%
positive indices.
6.2 according to 4B06]
Solve the equation 4M%0 = 8.
and express your answer with
positive indices.
&$ &#
Simplify
HKCEE 2003 Q4 [This should belong to section
HKCEE 1997 Q2(a)
Simplify
HKCEE 2002 Q1
and express your answer with
HKCEE 2004 Q1
$
Simplify
O& .& (P
('
and express your answer with
positive indices.
positive indices.
10
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 2005 Q2
Simplify
OM $ NP
HKDSE 2012 Q1
'
and express your answer with
N%
Simplify
#.&' $2
and express your answer with
$$
positive indices.
positive indices.
HKCEE 2006 Q1
HKDSE 2013 Q1
Simplify
O& $P
M '3 N &$
%
and express your answer with
& ./
Simplify (M % N)/ and express your answer with
positive indices.
positive indices.
HKDSE 2014 Q1
HKCEE 2007 Q2
#/
OMN .' P
$
Simplify #0 $.% and express your answer with
Simplify
positive indices.
positive indices.
HKCEE 2008 Q1
HKDSE 2015 Q1
Simplify
(&()$
&'
and express your answer with
and express your answer with
N#
#0
Simplify (#$ $.1)% and express your answer
positive indices.
with positive indices.
HKCEE 2009 Q1
HKDSE 2016 Q1
M'
Simplify (M .1
N)$
and express your answer with
Simplify
OM 2 N 1 P
'
and express your answer with
M % N ./
positive indices.
positive indices.
HKCEE 2010 Q1
HKDSE 2017 Q2
($
7
$
O## $.& P
Simplify -02 0&' 1 and express your answer
Simplify
with positive indices.
with positive indices.
HKCEE 2011 Q2
HKDSE 2018 Q2
M /%
(#.' )%
MN 1
and express your answer
Simplify (M # N $ )' and express your answer with
Simplify (M .'
positive indices.
with positive indices.
HKDSE SP Q1
HKDSE 2020 Q1
Simplify
(MN)'
M .% N /
and express your answer with
Simplify
N $ )#
O#$.' P
%
and express your answer with
#.#
positive indices.
positive indices.
HKDSE PP Q1
HKDSE 2021 Q1
Simplify
/
O#% $.' P
## $.$
and express your answer
with positive indices.
and express your answer
Simplify (TU! )(T '1 U2 )7 and express your
answer with positive indices.
HKDSE 2022 Q1
Simplify
O& $ ( .' P
& .% ( /
#
and express your answer
with positive indices.
11
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Past Paper (Paper 2) [Occurrence: High]
HKCEE SP Q36
1
HKCEE 1972 Q15
Which of the following is identical to
&
4 . 5 '
I J $I J
5
4
' '
.
= $S$
?
1
$
=
27 '!
F G =
64
A.
!
B.
2
A. 0S 1
C.
B. (VW)1
D.
%
C. 31'$ .
D. 3$'1 .
0
1
E. 3!$%1 .
.
HKCEE 1978 Q6
0
C. − 2.
0
2
(8 $%0 )1
=
8 1$'0
.
A. 8 1 .
E. 4.
B. 8 ! .
C. 8 2 .
HKCEE 1977 Q13
0
/)1
0
/1
.
B. 31$ .
A. −2.
(- −
<
A. 31 .
=
D.
0"
3$%1
=
9$
HKCEE 1974 Q1
B.
<
0"
HKCEE SP Q37
= $
0S 1
E. 1
32
.
<
'
1
'7
!
.
E. − 0".
S $
C. 0=1
D.
2
D. 8 !'$ .
E. 8 $'! .
=
HKCEE 1978 Q45
&
'
1
A. (-/ − /1 ) .
X3&%( Y =
&
'
&
A. 3&%(%1 .
B. 0( − 11 .
&
B. 3&
&
'
C. 0(1 − 1.
' %( '
.
'
C. 3(&%() .
&
&
D. −/' (- − /)' .
&
E. /1 (- − /)' .
D. 31&%1( .
E. 9&
' %( '
12
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 1982 Q2
HKCEE 1980 Q2
125& ⋅ 5( =
A. 625&%( .
B. 625&( .
C. 125&%!( .
D. 5&%!( .
E. 5!&%( .
HKCEE 1980 Q8
5$%1 − 35(5$'0 )
=
18(5$%0 )
A.
B.
C.
0
0L
0
07
0
7
.
81M ⋅ 4!M
=
2M ⋅ 161M
A. 2!M .
B. 21M .
C. 2M .
D. 8.
E. 1.
HKCEE 1983 Q4
1
0
(8 1 S '0 ) ÷ F8 1 S '0 G =
A. 8S.
B. 8S '0 .
.
C. 8S '! .
&
.
D. 8 1 S ' .
&
D. 5.
E. 8 '' S '1 .
E. 5$ .
HKCEE 1984 Q3
HKCEE 1981 Q1
(-1 /'! )1
=
-'1 /
(2$%0 )1 × (2'1$'0 ) ÷ 4$ =
A. 1.
A. -1 /'K .
B. 21$'0 .
C. 2$
' %1$
.
C. -" /'1 .
D. 2$
' '1$
.
D. -" /'" .
E. 2'1$%0 .
1 '7
B. - /
.
E. -" /'K .
HKCEE 1988 Q1
HKCEE 1981 Q4
(2M )M =
2$%2 − 2(2$ )
=
2(2$%! )
A.
K
B. 2M ⋅ 8 M .
B.
K
C. 28 M .
C. 1 − 2$%0 .
D. 21M .
D. 2$%2 − L.
6
A. 2M .
'
E. 2M .
L
2
.
.
0
E. 2$%0 .
13
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 1989 Q1
3$'0 × 3$%0 =
A. 3$
' '0
$' '0
B. 9
.
.
HKCEE 1992 Q8
" 89:;<
UV
VVWVVVX
$×$×⋯×$
Simplify $%$%⋯%$.
YZZZ[ZZZ\
" 89:;<
A. 7
$'1
"
C. 31$ .
B. 7 '
D. 61$ .
C. 7 − 2
E. 91$ .
D.
$
1
E. 1
HKCEE 1989 Q3
HKCEE 1994 Q33
8
[ =
√8
(3M )1 =
A. 3M
B. 3M%1
$
A. 8 # .
C. 31M
&
#
B. 8 .
D. 6M
&
'
C. 8 .
E. 91M
&
HKCEE 1995 Q4
D. 8 '# .
$
E. 8 '# .
&/
Simplify 0(&'1
HKCEE 1990 Q1
A.
B.
(-1$ )! =
A. -"$ .
C.
B. -L$ .
C. -
1$$
D. -
"$$
D.
.
E.
.
'
$
(2
&#
( &2
&0
&#
(2
&0
( &2
0
& # ( &'
27M
=
3N
HKCEE 1991 Q1
(-1& )(3-2& )
'
HKCEE 1996 Q2
$
E. -L$ .
A. 3-
'
=
"&
A.
<M
N
.
6
B. 9= .
B. (3-)"&
C. 9M'N .
C. 3-L&
D. 3 = .
D. 4-"&
E. 3!M'N .
$6
E. (32& )(-"& )
14
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 1998 Q7
HKCEE 2003 Q4
(2# )1
=
8#
3M ⋅ 9N =
A.
1
!
A. 3M%1N .
.
B. 3M%!N .
B. 2'# .
C. 27M%N .
C. 2# .
D. 27MN .
D. 2#
' '!#
E. 21#
.
' '!#
HKCEE 2004 Q1
.
21$ ⋅ 9$
=
3$
HKCEE 2000 Q3
A. 61$ .
(-! /'0 )'1
=
(-'0 /1 )2
A.
B.
C.
D.
E.
B. 6!$ .
C. 12$
0
&( $
0
&' ($
0
&' (/
0
&' (0
&#
(/
D. 121$ .
.
.
HKCEE 2005 Q1
.
- ⋅ -(- + -) =
A. -2 .
.
B. 2-! .
C. -! + -.
HKCEE 2001 Q10
-$'1
+
-$'0
-$'1
D. 3-1 + -.
=
HKCEE 2006 Q1
A. -$'0 .
(28)! ⋅ 8 ! =
B. -$'1 (1 + -).
A. 68 " .
C. 1 + -$'0 .
B. 88 " .
0
D. 1 + .
&
C. 68 < .
E. 1 + -.
D. 88 < .
HKCEE 2002 Q3
24.123
2M ⋅ 8N =
0B. 2
A. 2M%!N .
!MN
C. 16
.
M%N
D. 16MN .
.
y
2x 237
2 34
HKCEE 2007 Q1
If 7 is a positive integer, then 31$ ⋅ 4$ =
A. 61$ .
OB.
6!$ .
C. 121$ .
D. 12!$ .
32 22h
3 2
62h
15
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 2008 Q1
HKCEE 2011 Q1
1 LLL
F G (−2)LLK =
2
1 !!!
5!!2 F− G
=
5
A. −2.
A. −5.
B. −0.5.
B. −0.2.
C. 0.
C. 0.
D. 0.5.
D. 5.
HKCEE 2009 Q1
HKDSE SP Q1
2$ ⋅ 3$ =
2
A. 5$ .
2
OC. 8 .
B. 6$ .
$
(3-)1 ⋅ -! =
3
A. 3-7 .
3
B. 6-" .
C. 9-7 .
6
D. 9$ .
D. 9-" .
HKDSE 2012 Q1
(28 2 )!
=
28 7
HKCEE 2010 Q2
1 7//
F G (37// )! =
9
A. 38 1 .
A. 0.
B. 38 K .
B. 37// .
C. 48 K .
C. 67// .
D. 48 7< .
HKDSE 2013 Q1
D. 187// .
(27 ⋅ 9$%0 )! =
A. 3"$%01 .
B. 3"$%07 .
HKCEE 2010 Q39
If - and / are positive numbers, then
√(
&
=
√&
÷
$
C. 3<$%01 .
D. 6<$%0L .
HKDSE 2014 Q1
A.
√(
.
&(
B.
√&(
.
(
C.
√&(
.
&(
D.
0
√& $ (
(
(27! )'7 =
A.
B.
.
C.
D.
0
!1$'
.
0
!1$&%
.
0
0/$&'%
0
0/$'#$
.
.
16
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKDSE 2015 Q2
HKDSE 2019 Q2
(3S " )2
=
3S 1
(68 K )1
=
48 7
A. 4S 7 .
A. 38 2 .
B. 4S L .
B. 98 2 .
C. 27S01 .
C. 38 < .
D. 27S 11 .
D. 98 < .
HKDSE 2020 Q1
68
=
(38 '7 )'1
HKDSE 2016 Q1
8111 ⋅ 5""" =
12312225666
A. 10""" .
LLL
B. 10
Iamb
2666.5666
.
C. 40""" .
2 51666
D. 40LLL .
0666
F 777 G 3222 =
9
555
A. 0.
C.
o
D.
!&&&
0
!'''
0
!///
!
72
M0
.
1
D.
.
!M 0
.
HKDSE 2021 Q1
1
B.
1M 2
B.
C.
HKDSE 2017 Q2
0
A. 548 L .
3444
(2$ )(8!$ )
=
64$
A. 4$ .
.
B. 41$ .
.
O
C. 4
.
D. 4'2$ .
'!$
81$%0
=
41$%0
A. 1
24h
.
811$%!
É
(27$%0 )1
=
O
C. 3
B. 31$%" .
2$%L
C. 2$
42
134124
A. 3.
B. 2
2h
22
HKDSE 2022 Q2
HKDSE 2018 Q1
1
2.2
.
3
at
D. 30/$%02 .
D. 2'$
324
Chl
26 8
0
3
4
6
17
0
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
6.2 Exponential Equations
Definition:
An exponential equation is an equation with unknown in the index. For example,
-
2! = 8
-
4"!#$ = 64
-
%!&'
2
(!#'
=4
2
43
23
2
x 3a
3 2 3
43
24 1 210
2
4 1
x
10 2
4
121
Property:
For . > 0 and . ≠ 1, if . ! = . ) , then 2 = 3.
Simply speaking, if the bases are the same, we can compare their indices directly.
Example 6.5 (Refer to Example 6.5 from 4B06)
Solve the exponential equation 41M = √8.
Solution:
General approach:
(1) Convert the LHS and RHS to the same base.
(2) Compare their indices
41M = √8
0
Both 4 and 8 and be expressed
as powers of 2.
(21 )1M = (2! )1
!
22M = 21
48 =
8=
3
2
3
8
Exercise 6.5a
0
Solve the exponential equation 9M = ?1K.
34 3 2
2x
x
32
Ey y
E
y
y's
1337
3
3
18
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.5b
Solve the exponential equation 16
5
3
-x
=8 .
23
1245
2M 25
4
25
x
541
Exercise 6.5c
Solve the exponential equation 3
3x
•
9
x +1
w
33
5
-2 x
3
.
2
32
33
Combine
æ1ö
=ç ÷
è 81 ø
3
Xt 2 2
2
4
4
fx
38X
8X
x
1
19
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Example 6.6 (Refer to Example 6.6 from 4B06)
Solve the exponential equation 3M%0 − 3M = 18.
Solution:
3M%0 − 3M = 18
3
M (3
You cannot compare when you
have “+” or “-” on the LHS.
− 1) = 18
Therefore, you need to simplify
the LHS to compare. You can
do it by taking out common
factors
3M (2) = 18
3M = 9
3M = 31
8=2
Exercise 6.6a
Solve the exponential equation 2M + 2M'0 = 48.
2
ITE
3
2
2
1
4
48
21
48
2X
32
4
51
Exercise 6.6b
x -1
x
Solve the exponential equation 5 + 5 = 150 .
20
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.6c
x+2 y
ì
=2
ï4
.
x+ y
ï
3
=
1
î
Solve the simultaneous equations í
244
3
2
5 1
or
1 30
22 4
2
zxty
30
3
0
atty
xty
o
Exercise 6.6d
0
2M%N − 0" = 0
Solve the simultaneous equations ]
.
4M'N'1 − 1 = 0
2
7
1
24
4 9
9
4
xty
x
x
y
1
2
2
4
40
4
2
y
3
21
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Selected Exercises:
Further Practice P.6.17
Exercise 6B
Level 1: Q3, Q6, Q7-14 Level 2: Q16, 18, Q20, Q21-22, Q23-24, Q26.
End of Section Exercises
Q1.
Solve the following exponential equations.
(a)
2 x -3 = 16
x
3
(b)
32 x =
x+2
1
81
= 36
2
3
(c)
5 = 25
(d)
6
(e)
9 x -1 = 27
(f)
8 - x = 32
(g)
( 7 ) 3 x = 49
(h)
16 2+ x = 64 x - 2
4
x-2
= 5(25 )
x
(i)
125
(k)
2
æ1ö
ç ÷ = 1-2 x
8
è4ø
(j)
27 x
= 81x - 2
2 x -1
9
(l)
163- 2 x ( 2 x ) =
x
Q2.
1
8x
Solve the following exponential equations.
(a)
2 x +1 + 2 x = 6
(b)
3 x - 3 x -1 = 54
(c)
4(4 x ) - 4 x -1 = 60
(d)
2 x +1 + 2 x -1 = 10
(e)
9 x - 2 + 32 x = 82
(f)
2(4 x + 2 ) - 9(4 x ) = 92
(g)
9(2 ) - 4 =
(h)
6 x +1 - 6 x + 6 x -1 = 186
(i)
ì2 x - y = 8 y
ï
íæ 1 ö 2 y -1
= 3 x -5
ïç ÷
îè 3 ø
(j)
ìï9 y = 33 x -1
í x
ïî16 - 4 y - x = 0
x
x
2
1
2
22
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
6.3
Exponential Functions and Their Graphs
6.3.1 Exponential Functions
Definition:
An exponential function is a function of the form 3 = . ! or 4(2) = . ! , where . > 0 and
. ≠ 1. We call 3 = . ! an exponential function with base 7.
Q:
Why - ≠ 1?
A:
If - = 1, then S = 1M = 1 for every real number 8. Therefore, it is no longer an exponential
function.
Q:
Why - > 0?
A:
If we allow - < 0, say, - = −4, then S = (−4) M is undefined for some values of 8. When
&
0
8 = 1, then (−4)' = √−4 is NOT a real number.
Example:
Let S = 4M . Find the values of S when
(a)
a
7
(b)
8=1
0
(c)
8=1
8 = −1.5
x
is
II
y y't
9 4
4
4
y
15
L
Exercise:
Let `(8) = 9'M . Find the values of
(a)
a
(b)
`(−2)
firs 9
t
811
!
`0 1
1
(c)
`(−2.5)
b 5131 9
1
Cafe 2.5
942J
2431
23
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Example 6.7 (Refer to Example 6.7 from 4B06)
Given that `(8) = 3(4M ) and a(8) = 0.7M , find the values of the following expressions.
(a)
(b) `(2.5) × a(2.5)
`(0) − a(1)
(Give your answers correct to 3 significant figures if necessary.)
Solution:
(a)
Recall that -/ = 1 and -0 = - for every nonzero real number -.
`(0) − a(1) = 3(4/ ) − 0.70
= 3(1) − 0.7
= 2.3
(b)
`(2.5) × a(2.5) = 3(41.7 ) × 0.71.7
= 39.4 (3 s. f. )
Exercise 6.7a
0 M
Given that `(8) = 071 and V(8) = 5(0.6M ), find the values of the following expressions.
(a)
(b)
`(−1) + V(1)
`(1.8) ÷ V(1.8)
(Give your answers correct to 3 significant figures if necessary.)
a
1474 pls
E
t
b
fit
I
510.6
81
8
Pll
t
0.0277
Exercise 6.7b
f
510 6318
3
S.f
2x
x
Given that f ( x) =
and g ( x) = 1.8 , find the values of the following expressions.
5
(a)
` (1) + a(0)
(b)
` (1.5) × a(-1)
(Give your answers correct to 3 significant figures if necessary.)
24
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Example 6.8 (Refer to Example 6.8 from 4B06)
The population (d) of a town e years after the beginning of 2010 is estimated by
d(e) = 35000(1.0025)5 .
(a) Find the estimated population of the town
(i)
at the beginning of 2010,
(ii)
at the beginning of 2020.
(b) Will the increase in the estimated population of the town from the beginning of 2019 to the
beginning of 2020 exceed 90? Explain your answer.
(Give your answers correct to 3 significant figures if necessary.)
Solution:
Remark:
The notation d(e) means d is a function of e, i.e., the change in population depends on the
change in time.
(a)
(i)
At the beginning of 2010 è e = 0.
d(0) = 35000(1.0025)/
= 35000
(ii)
At the beginning of 2020 (10 year after the beginning of 2010) è e = 10.
d(10) = 35000(1.0025)0/
= 35885 (3 s. f. )
(b)
35900
3s.f
The increase is given by d(10) − d(9), i.e., the population of 2020 – the population of 2019:
d(10) − d(9) = 35000(1.0025)0/ − 35000(1.0025)<
≈ 89.4884
< 90
∴ The increase in the estimated population of the town from the beginning of 2019 to the
beginning of 2020 will not exceed 90.
25
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.8a
The number (h) of words that Jason can remember after studying a passage for e minutes is
estimated by h(e) = 50(1 − 0.95 ).
(a)
Find the number of words that Jason can remember after studying for 5 minutes, correct to
the nearest integer.
(b)
State whether the percentage increase in the number of words that Jason can remember in
the 6th minutes of study exceeds 12%. Explain your answer.
26
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.8b
At the beginning of a certain year, Peter deposits $10 000 into a bank at an interest rate 4% p.a.
compounded half-yearly. The interest ($I) received at the end of the nth year is given by
51 1$
i(e) = 10000 jF G − 1k
50
En
(a)
Find the interest received at the end of the 10th year.
(b)
Will the increase in the interest from the end of the 10th year to the end of the 15th year
exceed $3000? Explain your answer.
(Give your answers correct to the nearest dollar if necessary.)
a
1110
10000
4859
Eo
1
47396
4859
The interest received
b
2115
1110
4859
is
10000
f
10000
3254.14
The
increase
will exceed
1
Eo
g
73000
3000
27
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
y
6.3.2 Graphs of Exponential Functions
First case: the graph of S = - M when - > 1.
RHS
The figure below shows the graph of S = 2M and S = 3M .
Observation:
When ' > 0, 3> > 2>
When ' < 0, 3> < 2>
24 2,44
113
9
2,3 53 Tl
a
b
a
b
l
RHS
24 3 244
1 2
t s
4
a
2
i
11
(1) The graph must pass through (l, m), because when 8 = 0, S = -/ = 1.
(2) The graph will not intersect the n-axis, i.e., - M will remain positive for any real number 8.
0
Because even when 8 < 0, e.g., 8 = −2, S = -'1 = &' is still positive.
y L
Second case: the graph of S = - M when 0 < - < 1.
0 M
0 M
The figure below shows the graph of S = 011 and S = 0!1 .
2 X
a
x
y
t 0.7
yet
4
b
2
LHS
Ocaibcl
b at
a b
at
(3) The graph must pass through (l, m), because when 8 = 0, S = -/ = 1.
(4) The graph will not intersect the n-axis, i.e., - M will remain positive for any real number 8.
The reason is the same as the first case.
Remark:
The case S = - M when 0 < - < 1 is the same as the case S = -'M when - > 1.
28
Fr X
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
In summary
Range of Graph of ! = # !
Common
features
- > 1
0 < - < 1
1.
The graph cuts the S-axis at (0, 1).
2.
The graph never cuts the 8-axis. It lies above the 8-axis.
3.
The graph has neither a maximum point, a minimum point nor an
axis of symmetry.
Differences
1. The value of S increases as 8
increases.
2. The value of S gets closer and
closer to zero as 8 decreases
indefinitely.
3. As 8 increases, the rate of
increase of S becomes greater.
1. The value of S decreases as 8
increases.
2. The value of S gets closer and
closer to zero as 8 increases
indefinitely.
3. As 8 increases, the rate of
decrease of S becomes smaller.
Remark:
# !
The graphs of S = - ! and S = -"! (or S = 0$1 ) when - > 1 are symmetric with respect to the Saxis, for example:
# !
Note that, we will be using S = -"! and S = 0 1 to denote the case of S = - ! when 0 < - < 1.
$
They are the same because
1 !
1
F G = ! = -"!
-
29
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Example 6.9 (Refer to Example 6.9 from 4B06)
The figure shows the graph of S = 5! .
(a)
(b)
Using the graph, find the values of
(i)
√125,
(ii)
0%1
# &.(
.
Solve 5! = 20 graphically.
Solution:
(a)
To look at the graph, we must convert the expression in the form of 5! first.
(i)
√125:
√125 = N5)
)
= 5*
)
Therefore, we should look at the value of S = 5! when 8 = *, i.e.,
Hence, by looking at the graph, √125 ≈ 11.0.
30
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
(ii)
# &.(
0% 1
:
1 &.(
F G = 5"&.(
5
Therefore, we should look at the value of S = 5! when 8 = −0.4, i.e
# &.(
Hence, by looking at the graph, 0%1
(b)
≈ 0.5.
To solve 5! = 20, we look at the graph from a different direction.
We already know S = 50 when 8 equals to some unknown value, therefore, we should look
at the graph in the following way:
Therefore, 8 ≈ 1.9. That is, when 8 = 1.9, then S = 5#.+ ≈ 20.
31
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Exercise 6.9a
Using the graph shown in Example 6.9,
(a)
find the values of
(i)
(b)
5
25 ,
(ii)
æ1ö
ç ÷
è5ø
-1.8
,
x
solve 5 = 15 graphically.
Exercise 6.9b
The figure shows the graph of S = 0.5! .
(a)
Using the graph, find the values of
(i)
(b)
)
X√0.5Y ,
(ii)
√2.
Solve 0.5! = 6 graphically.
32
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Selected Exercises from Textbook:
Further Practice: P.6.27
Exercise 6C:
Level 1: Q2, Q4, Q8, Q9-10, Q13
Level 2: Q16, Q20, Q22, Q24, Q26
End of Section Exercise:
0.5 x
Q1. Given that f ( x) = 2(3 ) and g ( x) =
, find the values of the following expressions.
5
x
(Give your answers correct to 3 significant figures if necessary.)
(a) f (1) + g (1)
æ3ö
è7ø
(b) 2 f (0) - g (-1)
(c) f (1.5) + 3 g (-0.5)
x
Q2. If f ( x) = aç ÷ and f (2) = 9 , find the value of -.
æ 5t ö
Q3. The growth in the hair length (L mm) of a person after t days is estimated by L = 0.02çç1.3 ÷÷ .
è
ø
Find the growth in the hair length of the person after 150 days.
(Give your answer correct to 3 significant figures.)
x
æ3ö
Q4. The following figure shows the graphs of the exponential functions y = 7 and y = ç ÷ ,
è2ø
x
x
x
æ1ö
æ2ö
sketch the graphs of y = ç ÷ and y = ç ÷ on the same coordinate plane.
è7ø
è3ø
33
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Q5. The following figure shows the graphs of the exponential functions y = 0.2 x and y = 0.5 x ,
sketch the graphs of y = 2 x and y = 5 x on the same coordinate plane.
Q6. The figure shows the graph of y = 3 x .
(a)
Using the graph, find the values of
(i)
(b)
1.7
3 ,
(ii)
æ1ö
ç ÷
è3ø
-0.8
.
Solve the following equations graphically.
(i)
3 x = 10
(ii)
3 x = 24
Q7. It is given that f ( x) = 0.25(k x ) (where k > 0 ) and f (2) = 25. Find the values of
(a)
k,
(b)
100 f (0.5)
.
f (3)
(Leave your answers in surd form if necessary.)
x
æ4ö
Q8. It is given that f ( x) = 2(3 ) and g ( x) = ç ÷ .
è5ø
x
(a)
Find the values of the following expressions.
(i)
(b)
5 f (2) • 5 g (2)
(ii)
[ f (0.5)]2
g (0.5)
(Leave your answers in surd form if necessary.)
If 2 f (k ) • g (-1) = 45 , find the value of k.
34
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Q9. It is given that p ( x) =
10
, q ( x) = 1.6 x and r ( x) = p ( x) • q ( x) .
x
0.2
(a)
Express r(x) in the form kax, where a and k are constants.
(b)
Find the values of the following expressions.
(i)
æ1ö
rç ÷
è3ø
(ii)
r (-2)
Q10. The population (P) of a city after t years is estimated by P = 1 500 000(1.48t ).
(a)
Find the estimated population of the city this year.
(b)
Find the increase in the estimated population of the city in the 3rd year, correct to the
nearest thousand.
Q11. The total number (N) of cups of coffee sold by a coffee shop in a month is given by
N = C (1.005 x ) - 80 , where C is a constant and x (in dollars) is the water bill of that month.
Suppose no cups of coffee can be sold when there is no water.
(a)
Find the value of C.
(b)
If the water bill of the coffee shop was $900 last month, find the number of cups of
coffee sold last month. (Give your answer correct to the nearest integer.)
Q12. John invests $P in a certain bond at an interest rate 6% p.a. compounded yearly. The
amount ($A) he can get after n years is given by:
A = P(1 + 6%) n
Suppose John can get an amount of $22 472 after 2 years.
(a)
Find the value of P.
(b)
(i)
Could John get an amount of $2P after 10 years? Explain your answer.
(ii)
If the interest rate is doubled, will the interest got by John be more than $P after
7 years? Explain your answer.
35
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
x
æ2ö
Q13. The figure shows the graph of y = ç ÷ .
è5ø
(a)
Using the graph, find the values of
æ2ö
ç ÷
è5ø
(i)
(b)
5
-0.4
,
(ii)
æ 5 ö2
ç ÷ .
è2ø
Solve the following equations graphically.
x
æ2ö
ç ÷ =4
è5ø
(i)
(ii)
0.4 x = 7
Q14. The figure shows the graph of y = 6x.
(a)
Using the graph, find the
values of
æ1ö
è6ø
-0.8
(i) ç ÷
,
(ii) 216 .
(b)
Solve
6! − 30 = 0
graphically.
(c)
It is given that the graphs of
S = `(8) and S = 6! have
reflectional symmetry about
the S-axis.
(i) Sketch the graph of S =
`(8) on the same graph.
(ii) Find the algebraic
representation of the
function f(x).
36
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
Past Paper (Occurrence: Low-Medium)
HKCEE 2006 Q37
HKCEE 2009 Q39
The figure shows the graph of S = 4! . The
Which of the following may represent the
coordinates of d are
graph of S = −3! ?
A.
A. (1,0).
B. (0,1).
C. (4,0).
B.
D. (0,4).
HKCEE 2008 Q38
The figure shows the graph of S = - ! , the
graph of S = / ! and the graph of S = p ! on
C.
the same rectangular coordinate system,
where -, / and p are positive constants.
Which of the following must be true?
b l
b
occE
a
a
I
D.
acb
(1) - > /
X
(2) / > p
(3) - > 1
(4) p > 1
X
A. (1) and (3) only
P 6.41
42
P 6.45
B. (1) and (4) only
C. (2) and (3) only
Of
D. (2) and (4) only
37
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKCEE 2010 Q38
HKDSE 2014 Q32 [Concepts from 4B07 involved.]
The figure shows the graph of S = - ! + /,
The figure shows the graph of S = / ! and the
where - and / are constants. Which of the
graph of S = p ! on the same rectangular
following must be true?
coordinate system, where / and p are positive
constants. If a horizontal line r cuts the S-axis,
the graph of S = / ! and the graph of S = p !
at s, t and u respectively, which of the
following are true?
A. 0 < - < 1 and / > 0
B. 0 < - < 1 and / < 0
C. - > 1 and / > 0
D. - > 1 and / < 0
(1) / < p
(2) /p > 1
(3)
,,.
= log / p
A. (1) and (2) only
B. (1) and (3) only
HKCEE 2011 Q38
The graph shows the graph of S = 7"! . The
C. (2) and (3) only
D. (1), (2) and (3)
coordinates of q are
A. (1,0).
B. (0,1).
C. (7,0).
D. (0,7).
38
St. Catharine’s School for Girls | HKDSE – Exponential Functions | Notes
HKDSE 2020 Q33
The figure shows the graph of S = - ! and the graph of S = / ! on the same rectangular coordinate
system, where - and / are positive constants. If the graph of S = - ! is the reflection image of the
graph of S = / ! with respect to the S-axis, which of the following are true?
I.
-<1
II.
/>1
III.
-/ = 1
A. I and II only
B. I and III only
C. II and III only
D. I, II and III only
39