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