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. Revision Heinemann Topic Unit Sub skills Pack Textbook Questions

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

14 -15

15I, 15J

14 -15

15I, 15J

11 - 13

15H

 Solve for a and b equations of the following forms, given two pairs of corresponding values of x and y :

log y  b log x  log a , y  ax b and,

log y  x log b  log a, y  ab x  Using a straight line graph to confirm b x relationships in the form y  ax and y  ab

 Model mathematically situations involving the logarithmic or exponential function

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 C

Question should be completed without a calculator. 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. JGHS – H – EF1.1 Revision

FORMULAE LIST Circle: The equation x2  y 2  2 gx  2 fy  c  0 represents a circle centre  g ,  f  and radius g2  f 2  c .

The equation x  a 2   y  b2  r 2 represents a circle centre a, b  and radius r .

Scalar Product:

a.b  a b cos  , where  is the angle between a and b

or

 a1   b1      a.b  a1b1  a2 b2  a3b3 where a   a 2  and b   b2  . a  b   3  3

Trigonometric formulae:

sin  A  B   sin A cos B  cos A sin B

cos  A  B   cos A cos B  sin A sin B sin 2 A  2 sin A cos A cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2 sin 2 A

Table of standard derivatives:

Table of standard integrals:

f x 

f x 

sin ax

a cos ax

cos ax

 asin ax

f x 

 f x dx

sin ax

1  cos ax  C a

cos ax

1 sin ax  C a

JGHS – H – EF1.1 Revision

Q 1  NC

2  NC

Questions

Marks

Simplify the following expressions (a)

log 2 8

1

(b)

log 8 2  log 8 32

2

(c)

log 7 98  log 7 2

2

(d)

3 log 4 2  log 4 24  log 4 3

4

(e)

1 log 4 10  3 log 4 2  log 25 2

5

Given that

1 2 log a y  log a ( x  2)  3 log a 2 , show that y  64 x  2 2

3  NC

Solve the equation log 3 2 x  1  2

4  NC

Find the coordinates of where y  log 3 ( x  1)  2 cuts the x  axis.

5  NC

The graph y  log 4 ( x  3) passes through (q, 2) . Find the value of q

6  C

Solve the following equations to 3 significant figures

7  NC 8  NC

4

2

3

3

(a)

e 3 x1  4

3

(b)

ln 1  x   2

3

Solve 2 x  5 , leaving your answer in the form x 

Solve the equation 2 log x 6 

ln a where a and b are constants ln b

2 log x 8  2 where x  0 3

3

5

JGHS – H – EF1.1 Revision

9  NC

Solve log 3 6 x  log 3 ( x  2)  2

10  * NC

Functions f , g and h are defined on suitable domains by f ( x)  x 2  10 x  13 ,

11  C

12  C

4

g ( x)  2  x and h( x)  log 2 x . (a)

Find expressions for h f x  and hg x 

3

(b)

Hence solve h f x   hg x   3

8

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

N (t )  50e158t . (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

The population of a town is modelled by the equation Pt  Po e kt where P0 is the initial population, Pt is the population after time t years and k is the percentage growth rate. (a) (b) (c)

13  C

If town A has a population of 10 053 in 2007, what will the population be in 2021 with a growth rate of 1 6% ?

2

Town B had a population of 8 540 in 2010, what was their population in 1994 if the growth rate was 2  1% ?

3

Town C has a growth rate of 0  43% . How long would it take their population to triple?

4

 kt The Mass of a radioactive compound is given by the equation mt  mo e where m0 is

the initial mass, mt 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

JGHS – H – EF1.1 Revision

14  NC

The graph opposite has the relationship y  qx r . Find the values of q and r .

(4, 5) 3

5 15  NC

The graph opposite has the relationship y  ka x . Find the values of k and a .

2

6 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 y  a x 2 and y  a x  1 where a  1. (a) (b)

(c)

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

5

Show that the x -coordinate can be written in the form

x   log a  f (a)  log a g (a)

where f (a) and g (a) are functions of a .

3

a2 Show that the y -coordinate is y  2 a 1

2

[END OF REVISION QUESTIONS] [Go to next page for the Marking Scheme]

JGHS – H – EF1.1 Revision

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

1  NC

Marking Scheme

(a)

1

Answer

1

3

(b)

2

Use logarithm law

2

log 8 2  32   log 8 64

3

Simplify

3

2

4

Use logarithm law

4

(c)

log xy  log x  log y

x log  log x  log y y (d)

98  log 7 49 2

5

Simplify

5

2

6

Use logarithm law log x n  n log x

6

log 4 2 3  log 4 24  log 4 3

7

Use logarithm law

7

 2 3  24   log 4   3 

8

Start to simplify

8

log 4 64

9

Solution

9

3

10

Use logarithm law log x n  n log x

10

11

Use logarithm law

11

 3   10  2  log 4  1    25 2 

12

 10  8  log 4    5 

log xy  log x  log y and x log  log x  log y y

(e)

log 7

log xy  log x  log y and x log  log x  log y y

12

1 2

Know 25  5

log 4 10  log 4 2  log 4 25 3

13

Start to simplify

13

log 4 16

14

Solution

14

2

1 2

Notes: 1.

JGHS – H – EF1.1 Revision

2  NC

1

Use logarithm law log x n  n log x

1

2

Use logarithm law

2

3

Cancel log a and begin to simplify

3

4

Square both sides and finish

4

y  8( x  2) stated explicitly and y  64 ( x  2) 2 stated explicitly

log xy  log x  log y

1

log a y 2  log a ( x  2)  log a 2 3 1





log a y 2  log a 2 3 x  2 1

y 2  8( x  2) 2

Notes: 1. 3  NC

1

Start to solve

1

2 x  1  32

2

complete

2

x4

1

0  log 3 ( x  1)  2

Notes: 1. 4  NC

1

Set y  0 then start to solve

2  log 3 ( x  1) 2

Deal with log 3

2

3

Solve and state coordinates

3

32  x  1 x8

(8, 0) stated explicitly Notes: 1. Answer must be given in coordinate form 5  NC

1

Substitute point

1

2  log 4 (q  3)

2

Deal with log 4

2

42  q  3

3

solve

3

q  19

3x  1  ln 4

Notes: 1. 6  NC

(a)

(b)

1

Start to solve

1

2

Solve for x

2

3

Evaluate and round answer

3

0  129

4

Start to solve

4

1 x  e 2

5

Solve for x

5

x  1 e 2

6

Evaluate and round answer

6

 6  39

x

ln 4  1 3

Notes: JGHS – H – EF1.1 Revision

7  NC

1

Find ln of both sides

1

ln 2 x  ln 5

2

Use logarithm law log x n  n log x

2

x ln 2  ln 5

3

Solve for x

3

x

ln 5 ln 2

Notes: 2. 8  NC

1 2 3

Use logarithm law log x n  n log x 2 3

Know that 8  4 Use logarithm law

1 2 3

x log  log x  log y y

2

log x 6 2  log x 8 3  2 log x 36  log x 4  2

log x

36 2 4

4

Deal with log x

4

36  x2 4

5

solve

5

x3

Notes: 1. 9  NC

1

Use logarithm law

1

x log  log x  log y y

log 3

6x 2 x2

2

Deal with log 3

2

6x  32 x2

3

Start to solve

3

6 x  9( x  2)

4

solve

4

6 x  9 x  18 18  3 x x6

Notes: 1.

JGHS – H – EF1.1 Revision

10  * NC

(a)

(b)





1

Start process

1

h x 2  10 x  13

2

Complete for h f x 

2

log 2 x 2  10 x  13

3

Complete for hg x 

3

log 2 2  x 

4

Use logarithm law

4

 x 2  10 x  13    3 log 2  2x  

x log  log x  log y y





5

Convert to exponential form

5

x 2  10 x  13  23 2x

6

Process denominator

6

x 2  10 x  13  8(2  x)

7

Express in standard form

7

x 2 2 x  3  0

8

Solve for x

8

( x  1)( x  3)  0 x  1, 3

9

Disregard any unsuitable answers

9

x  1 (it should be clear that the answer x  3 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 2 (2  x) is x  2 11  C

(a)

1

State answer

1

50

(b)

2

Substitute

2

50 e1583

3

Answer (rounding not required)

3

5721

4

Interpret situation

4

4  e158t

5

Deal with e

5

ln 4  1 58t

6

Calculate t

6

7

Convert to minutes

7

(c)

t

ln 4  0  8774  1 58

Approximately 53 minutes.

Notes: 1.

JGHS – H – EF1.1 Revision

12  C

(a)

(b)

(c)

1

Substitution

1

Pt  10053  e 001614

2

Evaluate

2

12577 people in 2021

3

Substitution

3

8540  Po e 002116

4

Start to process

4

8540  Po e 002116 0336 e

5

Solve

5

Po  6102 people in 1994

6

Strategy

6

Eg P0  1 and Pt  3 or equivalent

7

Substitution

7

3  e 00043t

8

Resolve exponential

8

ln 3  0  0043 t

9

Solve

9

t

ln 3  255  5 years 0  0043

Notes: 1.

13  C

(a)

(b)

1

Substitution

1

245  300 e 3k

2

Start to solve and resolve exponential

2

 245  ln    3k  300 

3

Solve

3

 245  ln   300  k  3

4

Evaluate and round

4

0  067508   0  06751 to 4 sf



Strategy



Eg m0  1 and mt  0  5 or equivalent

6

Substitution

6

0  5  e 006751t

7

Resolve exponential

7

ln 0  5  0  06751t

8

Solve

8

5

5

t

ln 0  5  10  2673   0  06751

= 10 days 6 hours Notes: 1.

JGHS – H – EF1.1 Revision

14  NC

1

Take log 4 of both sides

1

log 4 y  log 4 qx r

2

Apply laws of logarithms

2

log 4 y  log 4 q  log 4 x r

3

Apply laws of logarithms

3

log 4 y  log 4 q  r log 4 x

4

Find r (by finding gradient)

4

r  gradient 

5

Find q (using y-intercept)

5

log 4 q  c

53 1  40 2

q  43 q  64 Notes: 1. 15  NC

1

Take log 8 of both sides

1

log 8 y  log 8 ka x

2

Apply laws of logarithms

2

log 8 y  log 8 k  log 8 a x

3

Apply laws of logarithms

3

log 8 y  log 8 k  x log 8 a

log 8 a  gradient  4

Find a (by finding gradient)

4

log 8 a   a8

1 3 

1 3



1 8

5

Find k (using y-intercept)

5

20 1  06 3

1 3



1 3

8



1 2

log 8 k  c k  82 k  64

Notes: 1.

JGHS – H – EF1.1 Revision

16  NC

(a)

(b)

(c)

1

Equate functions

1

a x 2  a x  1

2

Collect like terms

2

a x 2  a x  1

3

Remove common factor

3

a x a2 1  1

4

Start to solve

4





ax 

1 a 1 2

5

solve

5

 1  x  log a  2   a  1

6

Factorise denominator

6

  1  x  log a   a  1a  1 

7

Apply laws of logarithms

7

x  log a 1 log a a  1  log a a  1

8

Know that log a 1 0

8

x   log a a  1  log a a  1

9

Substitute x into one of the equations

9

Simplify

10

10

Eg y  a

y

 1  loga  2   a 1 

1

1 1 a2 1 a2  1    a2 1 a2 1 a2 1 a2 1

Notes: 1. [END OF MARKING SCHEME]

JGHS – H – EF1.1 Revision