Higher Maths - EF1.1 Logarithms -...

JGHS – H – EF1.1 Revision

Higher Maths - EF1.1 Logarithms - Revision

This revision pack covers the skills at Unit Assessment and exam level for logarithms so you can evaluate

your learning of this outcome. It is important that you prepare for Unit Assessments but you should also

remember that the final exam is considerably more challenging, thus practice of exam content

throughout the course is essential for success. The SQA does not currently allow for the creation of

practice assessments that mirror the real assessments so you should make sure your knowledge covers

the sub skills listed below in order to achieve success in assessments as these revision packs may not

cover every possible question that could arise in an assessment.

Topic Unit Sub skills Revision

Pack Questions

Heinemann Textbook

Logarithms: Manipulating algebraic expressions

EF1.3

Simplifying an expression using the laws of

logarithms and exponents 1 - 2 15E, 15F

Solving logarithmic and exponential equations 3 - 9 15G

Solve for a and b equations of the following

forms, given two pairs of corresponding values

of x and y :

axby logloglog , baxy and,

,logloglog abxy xaby

14 -15 15I, 15J

Using a straight line graph to confirm

relationships in the form baxy and

xaby 14 -15 15I, 15J

Model mathematically situations involving the

logarithmic or exponential function 11 - 13 15H

When attempting a question, this key will give you additional important information.

Key Note

Question is at unit assessment level, a similar question could appear in a unit assessment or an exam.

Question is at exam level, a question of similar difficulty will only appear in an exam.

# The question includes a reasoning element and typically makes a question more challenging. Both the Unit Assessment and exam will have reasoning questions.

* If a star is placed beside one of the above symbols that indicates the question involves sub skills from previously learned topics. If you struggle with this question you should go back and review that topic, reference to the topic will be in the marking scheme.

NC Question should be completed without a calculator.

C Question should be completed with a calculator.

Questions in this pack will be ordered by sub skill and typically will start off easier within that subskill

and then get more challenging. Some questions may also cover several sub skills from this outcome or

even include sub skills from previously learned topics (denoted with a *). Questions are gathered from

multiple sources including ones we have created and from past papers. Extra challenge questions are for

extension and are not essential for either Unit Assessment or exam preparation.


FORMULAE LIST Circle:

The equation 02222 cfygxyx represents a circle centre fg , and radius

cfg 22 .

The equation 222rbyax represents a circle centre ba , and radius r .

Scalar Product: cos baa.b , where is the angle between a and b

or 332211 bababa a.b where

3

2

1

a

a

a

a and

3

2

1

b

b

b

b .

Trigonometric formulae:

A

A

AAA

AAA

BABABA

BABABA

2

2

22

21

12

2

22

sin

cos

sincoscos

cossinsin

sinsincoscoscos

sincoscossinsin

Table of standard derivatives: xf xf

axsin

axcos

axacos

axasin

Table of standard integrals: xf dxxf

axsin

axcos

Caxa

cos1

Caxa

sin1


Q Questions Marks

1

NC

Simplify the following expressions

(a) 82log 1

(b) 322 88 loglog 2

(c) 298 77 loglog 2

(d) 32423 444 logloglog 4

(e) 252

12310 44 logloglog 5

2

NC

Given that 2322

1aaa xy log)(loglog , show that 2264 xy

4

3

NC

Solve the equation 2123 xlog

2

4

NC

Find the coordinates of where 213 )(log xy cuts the x axis.

3

5

NC

The graph )(log 34 xy passes through ),( 2 q . Find the value of q

3

6

C

Solve the following equations to 3 significant figures

(a) 413 xe 3

(b) 21 xln 3

7

NC

Solve 52 x, leaving your answer in the form

b

ax

ln

ln where a and b are constants

3

8

NC

Solve the equation 283

262 xx loglog where 0x

5


9

NC

Solve 226 33 )(loglog xx

4

10

*

NC

Functions ,f g and h are defined on suitable domains by 13102 xxxf )( ,

xxg 2)( and xxh 2log)( .

(a) Find expressions for xfh and xgh 3

(b) Hence solve 3 xghxfh 8

11

C

The number of bacteria in a petri dish, )(tN , after t hours is modelled by the equation

tetN 58150 )( .

(a) How many bacteria where present initially? 1

(b) How many bacteria are present after 3 hours? 2

(c) How long will it take for the number of bacteria to quadruple? Give your answer

to the nearest minute. 4

12

C

The population of a town is modelled by the equation kt

ot ePP where 0P is the initial

population, tP is the population after time t years and k is the percentage growth

rate.

(a) If town A has a population of 10 053 in 2007, what will the population be in 2021

with a growth rate of %61 ? 2

(b) Town B had a population of 8 540 in 2010, what was their population in 1994 if

the growth rate was %12 ? 3

(c) Town C has a growth rate of %430 . How long would it take their population to

triple? 4

13

C

The Mass of a radioactive compound is given by the equation kt

ot emm where 0m is

the initial mass, tm is the mass after time t days and k is a constant.

(a) If the mass of the compound is 300g on Tuesday and by Friday it is 245g, calculate

the value of k to 4 significant figures 4

The half-life of a compound is defined by the time taken for the compounds mass to

half. Calculate the half-life of this compound. Give your answer in days and hours to the

nearest hour.

(c) Calculate the half-life of this compound 4


14

NC

The graph opposite has the relationship rqxy .

Find the values of q and r .

5

15

NC

The graph opposite has the relationship xkay .

Find the values of k and a .

5

16

NC

(although no individual part of this question is beyond the scope of the exam, it is

extremely challenging and even a strong A candidate may struggle to access it)

Two graphs are defined by 2 xay and 1 xay where 1a .

(a) Find the x -coordinate of the point of intersection of these two graphs in terms of

a . 5

(b) Show that the x -coordinate can be written in the form

)(log)(log agafx aa

where )(af and )(ag are functions of a . 3

(c) Show that the y -coordinate is 12

2

a

ay

2

[END OF REVISION QUESTIONS]

[Go to next page for the Marking Scheme]

3

(4, 5)

2

6


Where suitable, you should always follow through an error as you may still gain partial credit. If you are

unsure how to do this ask your teacher.

Q Marking Scheme

1

NC

(a) 1 Answer 1 3

(b) 2 Use logarithm law

yxxy logloglog

2 64322 88 loglog

3 Simplify 3 2

(c) 4 Use logarithm law

yxy

xlogloglog

4 49

2

9877 loglog

5 Simplify 5 2

(d) 6 Use logarithm law xnxn loglog 6 3242 44

3

4 logloglog

7 Use logarithm law

yxxy logloglog and

yxy

xlogloglog

7

3

2423

4log

8 Start to simplify 8 644log

9 Solution 9 3

(e) 10 Use logarithm law xnxn loglog 10 2

1

4

3

44 25210 logloglog

11 Use logarithm law

yxxy logloglog and

yxy

xlogloglog

11

2

1

3

4

25

210log

12

Know 525 2

1

12

5

8104log

13 Start to simplify 13 164log

14 Solution 14 2

Notes:

1.


2

NC

1 Use logarithm law xnxn loglog 1 32

1

22 aaa xy log)(loglog

2 Use logarithm law

yxxy logloglog

2

2232

1

xy aa loglog

3 Cancel alog and begin to simplify 3

)( 282

1

xy

4 Square both sides and finish 4 228 )( xy stated explicitly and 2264 )( xy stated explicitly

Notes:

1.

3

NC

1 Start to solve 1 2312 x

2 complete 2 4x

Notes:

1.

4

NC

1 Set 0y then start to solve 1

)(log

)(log

12

210

3

3

x

x

2 Deal with 3log 2 132 x

3 Solve and state coordinates 3 8x (8, 0) stated explicitly

Notes:

1. Answer must be given in coordinate form

5

NC

1 Substitute point 1 )(log 32 4 q

2 Deal with 4log 2 342 q

3 solve 3 19q

Notes:

1.

6

NC

(a) 1 Start to solve 1 413 lnx

2 Solve for x 2

3

14

lnx

3 Evaluate and round answer 3 1290

(b) 4 Start to solve 4 21 ex

5 Solve for x 5 21 ex

6 Evaluate and round answer 6 396

Notes:


7

NC

1 Find ln of both sides 1 52 lnln x

2 Use logarithm law xnxn loglog 2 52 lnln x

3 Solve for x 3

2

5

ln

lnx

Notes:

2.

8

NC

1 Use logarithm law xnxn loglog 1 286 3

2

2 xx loglog

2

Know that 48 3

2

2 2436 xx loglog

3 Use logarithm law

yxy

xlogloglog

3 2

4

36xlog

4 Deal with xlog 4 2

4

36x

5 solve 5 3x

Notes:

1.

9

NC

1 Use logarithm law

yxy

xlogloglog

1 2

2

63

x

xlog

2 Deal with 3log 2 23

2

6

x

x

3 Start to solve 3 )( 296 xx

4 solve 4

6

318

1896

x

x

xx

Notes:

1.


10

*

NC

(a) 1 Start process 1 13102 xxh

2 Complete for xfh 2 13102

2 xxlog

3 Complete for xgh 3 x22log

(b) 4 Use logarithm law

yxy

xlogloglog

4

32

13102

2

x

xxlog

5 Convert to exponential form 5 3

2

22

1310

x

xx

6 Process denominator 6 )( xxx 2813102

7 Express in standard form 7 0322 xx

8 Solve for x 8 031 ))(( xx

3 1,x

9 Disregard any unsuitable answers 9 1x (it should be clear that the

answer 3x has been eliminated, such as scored through, for this mark)

Notes:

1. This question includes the subskill relating to composite functions from EF1.3. If

you cannot remember how to do that, please revisit that topic.

2. For 9, remember that you cannot find the log of 0 or a negative number, therefore

the domain of )(log x22 is 2x

11

C

(a) 1 State answer 1 50

(b) 2 Substitute 2 358150 e

3 Answer (rounding not required) 3 5721

(c) 4 Interpret situation 4 te 5814

5 Deal with e 5 t5814 ln

6 Calculate t 6 87740

581

4

lnt

7 Convert to minutes 7 Approximately 53 minutes.

Notes:

1.


12

C

(a) 1 Substitution 1 14016010053 ePt

2 Evaluate 2 12577 people in 2021

(b) 3 Substitution 3 1602108540 ePo

4 Start to process 4 160210

3360

8540

eP

eo

5 Solve 5 6102oP people in 1994

(c) 6 Strategy 6 Eg 10 P and 3tP or equivalent

7 Substitution 7 te 004303

8 Resolve exponential 8 t004303 ln

9 Solve 9 5255

00430

3

lnt years

Notes:

1.

13

C

(a) 1 Substitution 1 ke 3300245

2 Start to solve and resolve exponential 2 k3

300

245

ln

3 Solve 3

3

300

245

ln

k

4 Evaluate and round 4 0675100675080 to 4 sf

(b) 5 Strategy 5 Eg 10 m and 50 tm or equivalent

6 Substitution 6 te 06751050

7 Resolve exponential 7 t06751050 ln

8 Solve 8 267310

067510

50

lnt

= 10 days 6 hours

Notes:

1.


14

NC

1 Take 4log of both sides 1 rqxy 44 loglog

2 Apply laws of logarithms 2 rxqy 444 logloglog

3 Apply laws of logarithms 3 xrqy 444 logloglog

4 Find r (by finding gradient) 4

2

1

04

35

gradientr

5 Find q (using y-intercept) 5

64

43

4

q

q

cqlog

Notes:

1.

15

NC

1 Take 8log of both sides 1 xkay 88 loglog

2 Apply laws of logarithms 2 xaky 888 logloglog

3 Apply laws of logarithms 3 axky 888 logloglog

4 Find a (by finding gradient) 4

2

1

8

1

8

18

3

1

3

1

60

02

3

3

13

1

8

8

a

a

gradienta

log

log

5 Find k (using y-intercept) 5

64

82

8

k

k

cklog

Notes:

1.


16

NC

(a) 1 Equate functions 1 12 xx aa

2 Collect like terms 2 12 xx aa

3 Remove common factor 3 112 aa x

4 Start to solve 4

1

12

a

a x

5 solve 5

1

12a

x alog

(b) 6 Factorise denominator 6

11

1

aax alog

7 Apply laws of logarithms 7 111 aax aaa logloglog

8 Know that 01alog 8 11 aax aa loglog

(c) 9 Substitute x into one of the equations

9

Eg 11

12

aa

aylog

10 Simplify 10

11

1

1

11

1

12

2

2

2

22

a

a

a

a

aay

Notes:

1.

[END OF MARKING SCHEME]

Higher Maths - EF1.1 Logarithms -...

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