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Maths Quest Maths B Year 12 for Queensland 2e 1
WorkSHEET 3.2 Exponential and logarithmic functions Name: ____________________________ 1 Solve for x in each of the following:
(a) ( ) 212log5 =-x (b) 34log36log2log 333 =-+x
(a) ( ) 212log 5 =-x
132622512512 2
===-=-
xx
xx
(b) 34log36log2log 333 =-+x
48366427
36436
36436log
34log6loglog
3
3
33
233
=
´=
=
=
=-+
x
x
x
xx
2 Solve for x in each of the following: (a) 216log144log =- xx (b) ( ) ( ) 12log4log 77 =++- xx
(a) 216log144log =- xx
39
216144log
2
==
=
xx
x
is the only possible solution (b) ( ) ( ) 12log4log 77 =++- xx
( )( )( )( )
( )( )50350152782724124log
2
2
17
==+-=--
=--
=+-
=+-
xxxxxxxxxxx
is the only possible solution. The logarithm of a negative number is not possible.
Maths Quest Maths B Year 12 for Queensland 2e 2
3 Solve for x in each of the following: (a) ( ) 032log12log 2
22 =+- xx
(b) 316log
log4
2
2 =x
(a) ( ) 032log12log 22
2 =+- xx
( )( )
256or162or2
8logor4log8or4
08403212loglet
8422
22
====
====
=--=+-
=
xxxx
xxaa
aaaa
x a
(b) 316log
log4
2
2 =x
( )
82loglog
2loglog
16log43log
16log3log4
322
43
422
22
22
==
=
=
=
xx
x
x
x
4 Solve for x in each of the following. Where necessary, give answers exact to three decimal places.
(a) x
x
ee
321 1=-
(b) 23 =-xe
(a) x
x
ee
321 1=-
1321
321
-=-=-
= --
xxx
ee xx
(b) 23 =-xe
405.032log
32loglog
32
=
÷øö
çèæ=-
÷øö
çèæ=
=
-
-
x
x
e
e
e
ex
e
x
5 Solvefor𝑥;
𝑒#$ − 𝑒$ − 6 = 0
Becomes; (𝑒$ + 2)(𝑒$ − 3) = 0
NFL𝑒$ = −2𝑜𝑟𝑒$ = 3
So,𝑥 = ln−2 𝑜𝑟 ln 3
clearlyonlyonesolution,
𝑥 = ln 3
Maths Quest Maths B Year 12 for Queensland 2e 3
6 Solve for x in each of the following. Give answers exact to three decimal places. (a) 056 =+- -xx ee (b) 35 <-xe ** just use an equal sign in this question **
(a) 056 =+- -xx ee multiply b.s. by xe
( )( )
0or609.11logor5log
150150562
====
==
=--
=+-
xxxx
eoreeeee
ee
xx
xx
xx
(b) 35 <-xe take elog of b.s.
901.33log5
3log53loglog 5
><-<-<-
xx
xe
e
e
ex
e
7 If kteNN -= 0 and N = 1000 when t = 2, and N = 2000 when t = 5, find values, correct to two decimal places, for 0N and k.
( )( )( ) ( )
96.6291000
1000
23.02log3132log
1222200011000
2log312
0
2log312
0
3
50
20
0
==
=
-=-=
-=÷=
=
=
=
-´
-´-
-
-
-
-
e
e
eN
eN
k
keeNeNeNN
e
e
k
k
k
kt
8 Solve the following by giving exact values for x: (a) ( ) 212log -=-xe (b) 6log4log3log eee x =-
(a) ( ) 212log -=-xe
( )121
1212
2
2
2
+=
+=
=-
-
-
-
ex
exex
(b) 6log4log3log eee x =-
8243
643
6log43log
==
=
=
xx
x
xee
Maths Quest Maths B Year 12 for Queensland 2e 4
9 Solve for x: ( ) ( ) 6log4log3log eee xx =-+-
( ) ( )( )( )
( )( )1or6
0160676127
6log43log6log4log3log
2
2
===--=+-
=+-
=--=-+-
xxxxxxxxxxxx
ee
eee
Only x = 6 is a possible solution as x = 1 leads to an impossible logarithm.
10 Express y in terms of x in: yx ee log12log4 =-
( )( )
( )
( )( )
( ) yex
yex
eyx
ey
x
y
x
yx
yx
e
ee
ee
=
=
=
=
=
=-
=-
2
8
214
21
4
1
21
4
21
4
21
4
2
2
2
2
12log
1log2log
log12log4
10 The population of tadpoles in a pond is given by: ( ) tetP 2.0300= where t is the time measured in weeks. (a) How many tadpoles were initially in the
pond?
(b) How many tadpoles were in the pond after 10 weeks?
(c) After what period of time will the population exceed 1000 tadpoles?
( ) tetP 2.0300= (a) when t = 0, P(t) = 300 (b) when t = 10, ( ) 2217300 2 == etP (c) when P(t) > 1000,
weeks6310log5
310log2.0
30010001000300
2.0
2.0
>
>
>
>
>
t
t
t
e
e
e
e
t
t