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Logs and ExponentialsHigher Mathematics Supplementary Resources

Section AThis section is designed to provide examples which develop routine skills necessary for completion of this section.

R1 I have had experience of simplifying expressions with surds and indices.

1. Simplify the following writing the answers with positive indices only.

(a) x2× x7 (b) y−3× y8 (c) x6÷ x4

(d) y−3÷ y−1 (e) (a4 )3 (f) ( p−4 )−2

2. (a) 4 x3×2 x2 (b) 5 x3×4 x−3 (c) 3x5 y3

6 x2 y5

(d) 4 r 8

2r−5 (e) √ x×x2 (f) √ x× 3√x2

(g) 1√a

×√a3 (h) 3√ x×x−13 (i) x−2 ( x2+1 )

R2 I can write an exponential in logarithmic form and vice versa.

1. For each exponential relationship, write a related logarithmic relationship.

(a) 3x=5 (b) 8t=100 (c) 5r=13

(d) 10r=5 (e) 6s=15 (f) 2p=32

Higher Mathematics – Logs and Exponentials

(g) ax=5 (h) b y=5 (i) c z=5

(j) 23=8 (k) 32=9 (l) 104=10000

(m) ab=c (n) x y=z (o) pq=r

2. For each logarithmic relationship, write a related exponential relationship.

(a) log x20=3 (b) log e x=2 (c) log37=x

(d) log y16=4 (e) log er=1 ∙2 (f) log69=t

(g) log3 x=2 (h) log10 y=2 ∙4 (i) log(e+1)R=v

(j) log(x+2)T=r (k) log5 r=x+1 (l) log e x=r−2

R3 I can solve exponential equations using logarithms.

1. Solve each of the following exponential equations

(a) 10x=1000 (b) 10x=0 ∙01 (c) 10x=100000

(d) 10x=20 (e) 10x=3000 (f) 10x=0 ∙05

(g) 102 p=5 (h) 109 t=500 (i) 10−4 r=0∙6

(j) 10−3 p=20 (k) 100 ∙ 1 y=3000 (l) 100 ∙ 5q=0 ∙05


(a) ex=7 (b) ex=23 (c) e2 t=9

(d) e3 p=16 (e) e−3 x=0 ∙4 (f) e−9 r=1∙3

(g) 6ex=12 (h) 4e0 ∙02 x=12 (i) 12e3 t=6

(j) 3e−4 p=21 (k) 2e−2x=0 ∙4 (l) 6e−0 ∙5 r=1 ∙3


(a) 4×10x=400 (b) 7×10p=0 ∙67 (c) 3×10t=12

SLC Education Resources - Calderglen High School

(d) 0 ∙5×10a=20 (e) 10b

8=12 ∙5 (f) 10c

100=0 ∙05


Section BThis section is designed to provide examples which develop Course Assessment level skills

NR1 I can manipulate logarithms and exponentials and apply the three main laws of logarithms.

1. Given b=e t which of the following is true:

(a) logt b=e

(b) log eb=t

2. Given log n x= y which of the following is true:

(a) n y=x

(b) x y=n

3. Simplify

(a) log x3+ logx 5−logx 7

(b) log a32−2 loga4

4. Show that (a) log38log32

¿3 . (b) logb9a2

logb3a ¿2 .

5. If log3 x=2 log3 y−3 log3 z find an expression for x in terms of y and z.

6. Find a if log a64=32 .

7. Simplify 2 loge (3e)−4 loge (2e ) expressing your answer in the form log eB−log eC−A where A ,B and C are whole numbers.


NR2 I can solve exponentials and logarithmic equations using the laws of logarithms.

1. Given the equation y=3×4x find the value of x when y=10 giving your answer to 3 significant figures.

2. Given the equation A=A0 e−kt, find, to 3 significant figures:

(a) A when A0=5, k=0 ∙23 and t=20.

(b) k when A=70, A0=35 and t=20.

(c) t when A=1000, A0=10 and k=−0.01.

3. (a) Solve the equation log3 (3−2x )+log3 (2+x )=1.

(b) Solve log3(5−x)−log3 (3−x )=2, x<3.

(c) Solve log 4 x+ log4 ( x+6 )=2.

4. (a) Given that log5 x=A, show that log25 x=12 A.

(b) Solve log 4 x+ log16 x=3 .

5. The curve with equation y=log4(x−1)−2, where x>1, cuts the x-axis at the point ( p ,0 ). Find the value of p.

6. If log 48+log4q=1, find the value of q.

7. Solve the equation log3(x+2)−3 log32=2.

8. Find x if 4 log x6−2 logx 4=1 .


2

1

9. Find the x-coordinate of the point where the graph of the curve with equation y= log3(x−2)+1 intersects the x-axis.

NR3 I have experience of plotting and extracting information from straight line graphs with logarithmic axes (axis).

1. Show that y=k xn, where k and n are constants, can be expressed as a straight line in terms of log y and log x.

2. Show that y=A ekx, where k and n are constants, can be expressed as a straight line in terms of log y and x.

3. Variables x and y are related by the equation y=k xn

The graph of log2 y against log2 x is a straight line through the points (0 ,3) and (4 ,5), as shown in the diagram.

Find the values of k and n.

4. Two variables, x and y, are connected by the law y=ax.

The graph of log e y against x is a straight line passing through the origin and the point A(8 ,4).

Find the value of a.

5.

The graph illustrates the law y=kxn.


P

Q

2∙1

-2

If the straight line passes through A(1 ,0) and B(0 ,2), find the values of k and n.

6. The result of an experiment gives rise to the graph shown.

(a) Write down the equation of the line in terms of Pand Q.

(b) If P=log10 p and Q=log10q, show that p and q satisfy p=aqb stating the values of a and b.

7. Variables x and y are related by the equation y=A ekx

The graph of log e y against x

is a straight line through the points (0 ,3) and (6 ,5), as shown in the diagram.

Find the values of k and A.

8. Two variables x and y satisfy the equation y=5×4x.


(a) Find the value of a if (a ,10) lies on the graph with equation y=5×4x.

(b) If (−12 , b) also lies on the graph, find b.

(c) A graph is drawn of log10 y against x. Show that its equations will be of the form log10 y=Px+Q and state the gradient of this line.


NR4 I can solve logarithmic and exponential equations in real life contexts.

1. The amount of a certain radioactive element, At , remaining after t years can be found using the formula At=A0 e

−0 ∙ 002t, where A0 is the amount present initially.

(a) If 300 grams are left after 500 years, how many grams were present initially.

(b) The half-life of a substance is the time taken for the amount to decrease to half its initial amount. What is the half life of this substance?

2. The value V (in £ thousand) of a car is shown to depreciate after t years from first purchase according to the formula V=18 e−0 ∙ 15t.

(a) What was the value of the car when first purchased?

(b) The car was sold when its value had dropped to 10% of the value when first purchased.

After how many years was the car sold?

3. The formula At=A0 e−0 ∙ 000124 t is used to determine the age of wood,

where A0 is the amount of carbon-14 in any living tree, At is the amount of carbon-14 in the wood being dated and t is the age of the wood in years.

A wooden artefact was found to contain 90% of the carbon-14 of a living tree.

Is the artefact over 500 years old?


4. The size of a rabbit population, N , can be modelled using the equation N=N 0 e

kt where N0 is the population at the beginning of a study and t is the time in years since the study began and k is a constant.

(a) The rabbit population comprised of 70 individuals at the beginning of the study. If k=0.05 find the size of the rabbit population after six years.

(b) How long will it take the rabbit population to double in size?

5. Radium decays exponentially and its half-life is 1600 years.

If A0 represents the amount of radium in a sample to start with and A( t) represents the amount remaining after t years, then (t)=A0 e

−kt .

(a) Determine the value of k, correct to 3 significant figures.

(b) Hence find what percentage, to the nearest whole number, of the original amount of radium will be remaining after 2500 years.

6. The concentration of a fertiliser in the soil can be modelled by the equation F=F0e

−kt where F0 is the initial concentration, F t is the concentration at time t and t is the time, in days, after the application of the pesticide.

(a) If it takes 20 days for the level of the fertiliser in the soil to reduce by 25%, find the value of k to 2 significant figures.

(b) Eighty days after the initial application, what is the percentage decrease in the concentration of the fertiliser?

7. The spread of disease in trees was described by a law of the form A( t)=A0 e

kt where A0 is the area covered by the disease when it was first detected and A is the area covered by the disease t months later.


If it takes six months for the area of the disease to double, find the value of the constant, k, correct to 3 significant figures.


NR5 I can display on, and extract information from, logarithmic and exponential graphs.

1. The diagram shows the curves with equations y=5x and y=4(1−x).

The graphs intersect at the point A.

(a) Show that the x-coordinate of A can be written in the form logaClogaD

, for all a>1.

(b) Calculate the y-coordinate of A.

2. The diagram shows the graph of y=px, p>1.

On separate diagrams sketch:

(a) y=−px

(b) y=p−x

(c) y=p2−x

3. Sketch the graph of y=loga x. On the same diagram sketch:

(a) y=loga( 1x )(b) y= loga(x−3)

(c) y=loga x+2


4. Sketch the graph of y=px+1.

(a) On the same diagram sketch the graph of y=px+1.

(b) Prove that the two graphs intersect at a point where the x-coordinate is log p( 1

p−1 ).

5. The diagram shows a sketch of part of the graph of y=log3x

(a) Write down the values of a and b.

(b) Sketch the graph of y=log3 (x−1 )+2

6. The function g is of the form g ( x )=log p(x−q).

The graph of y=g ( x ) is shown in the diagram.

(a) Write down the values of p and q.

(b) State the domain of g.


NR6 I have experience of cross topic exam standard questions

Logarithms with composite functions

1. Functions p, q and r are defined on suitable domains by

p ( x )=x2−12x+19, q ( x )=5−x and r ( x )=log2 x.

(a) Find expressions for r ( p ( x )) and r (q ( x )).

(b) Hence solve r ( p ( x ))−r (q ( x ) )=3.

Logarithms with polynomials

2. (a) Show that x=−4 is a root of x3+8 x2+11 x−20=0.

Hence factorise x3+8 x2+11 x−20 fully.

(b) Solve log2 ( x+3 )+log2¿¿.


AnswersR1

1. (a) x9 (b) y5 (c) x2

(d) 1y2

(e) a12 (f) p8

2. (a) 8 x5 (b) 20 (c) x3

2 y2

(d) 2 r13 (e) x52 (f) x

76

(g) a (h) 1 (i) (1+ 1x2 )R2

1. a) log35=x b) log 8100=t c) log513=r

d) log105=r e) log615=s f) log232=p

g) log a5=x h) log b5= y i) log c5=z

j) log28=3 k) log39=2 l) log10 10000=4

m) log ac=b n) log x z= y o) log p r=q

2. a) x3=20 b) e2=x c) 3x=7

d) y4=16 e) e1 ∙2=r f) 6t=9

g) 32=x h) 102 ∙ 4= y i) (e+1 )v=R

j) ( x+2 )r=T k) 5(x+1)=r l) e(r−2)=x

R3 (Answers to 3 significant figures where appropriate)

1. a) x=3 b) x=−2 c) x=5

d) x=1 ∙30 e) x=3 ∙48 f) x=−1 ∙30

g) p=0∙349 h) t=0 ∙300 i) r=0 ∙0555

j) p=−0 ∙434 k) y=34 ∙8 l) q=−2 ∙60


2. a) x=1 ∙95 b) x=3 ∙14 c) t=1 ∙10

d) p=0∙924 e) x=0 ∙305 f) r=−0 ∙0292

g) x=0 ∙693 h) x=54 ∙9 i) t=−0 ∙231

j) p=−0 ∙486 k) x=0 ∙805 l) r=3∙06

3. a) x=2 b) p=−1∙02 c) t=0 ∙602

d) a=1∙60 e) b=2 f) c=0∙699

NR1

1. (b) 2. (a) 3(a). log x (157 ) 3(b). log a2

4(a). Proof 4(b). Proof 5. x= y2

z36. a=16

7. log e9−loge16−2

NR2

1. x=0 ∙868 2(a). A=0 ∙0503 2(b). k=−0 ∙0347

2(c). t=461 3(a). x=−32 and x=1 3(b). 2 ∙75

3(c). x=−8 and x=2 4(a). Proof 4(b). x=16

5. p=17 6. q=12 7. x=70 8. x=81

9. x=2 13

NR3

1. log a y against log a x 2. log a y against log a x

3. n=12 and k=8 4. a=e

12

5. n=−2 and k=e2 6(a). P=1 ∙05Q+2∙1

6(b). a=126 and b=1∙057. k=13 and A=e3

8(a). a=0 ∙5 8(b). b=52 8(c). Proof and gradient = log10 4


NR4

1(a). 815 grams 1(b). 347 years 2(a). £18000 2(b). 15 ∙4 years

3. The artefact is 850 years old which is over 500 years old.

4(a). 94 rabbits 4(b). 13 ∙9 years 5(a). k=0 ∙000433

5(b). 34% 6(a). k=0 ∙014 6(b). 67% 7. k=0 ∙116

NR5

1(a).loga4loga20

1(b). y=2∙1

2(a). 2(b).

2(c).

3.


4(a).

4(b). Proof 5(a). a=1 and b=3

5(b).

6(a). p=4 and q=3 6(b). Domain = {x : x∈R , x>3 }

NR6

1(a). r ( p (x))=log2 (x2−12x+19 ) and r (q(x ))=log2 (5−x )

1(b). x=−3 and x=7


2(a). (x−1)(x+4)(x+5) 2(b). x=1 , x=−4 , x=−5


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