Exponential (Growth) functions and Logarithms

1

Objective Deadlines / Progress

Growth and logarithms

Know and use graphs of growth and logarithm functions. Sketch transformations of their graphs

Understand logarithms as inverses of growth functions and convert between

e.g.

𝟐^𝒙= 𝟕 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒕𝒐 𝒙 = 𝒍𝒐𝒈_𝟐𝟕

Use the laws of logarithms to rearrange and simplify equations and expressions

Solve equations of the form 𝑎

^𝑥

= 𝑏 Solve disguised quadratics such as e.g. 2

^2𝑥

− 7(2

^𝑥

) +

10

= 0

Convert a growth function to a linear function using logarithms and solve real life / modelling problems

Exponential and Natural logarithm

Know the exp function 𝑦 =

𝑒^𝑥 and natural logarithm function

𝑦 =

𝑙𝑛 𝑥

sketch

transformations of their graphs

Solve equations that are functions of

𝑒^𝑥 and 𝑙𝑛 𝑥

Understand why the gradient of 𝑦 =

𝑒^𝑥 Is ^𝑑𝑦

𝑑𝑥

=

𝑒^𝑥 and know the gradient function for

𝑦 =

𝑒^𝑎𝑥

Solve real life problems with functions of the

form 𝑃 = 𝐴

𝑒^𝑓(𝑥)

such as populations or

radioactive decay

2

WB1 Sketch a few examples of growth functions (exponential graphs)

Sketch at least three examples and describe the key features of the graphs

WB2 Sketch the graphs of 𝑦 = 2^𝑥and 𝑦 = 2^−𝑥 describe the key features of the graphs

3

WB3 Sketch a few examples of graphs of logarithm functions

Sketch at least three examples and describe the key features

WB4 Some equations involving indices can be solved by inspection. Try these Use your calculator to check

𝒂) 𝟐^𝒙= 𝟏𝟔

𝒃) 𝟏𝟎^𝒙= 𝟎. 𝟎𝟎𝟏

𝒄) 𝟓^𝒙= 𝟔𝟐𝟓

𝒅) (^𝟏_𝟑)^𝒙= 𝟖𝟏

𝒆) 𝟏𝟖(𝟐)^𝒙= 𝟗

𝒇) 𝟕(𝟗)^𝒙= 𝟐𝟏

𝒈) 𝒍𝒐𝒈_𝟏𝟎𝒙 = 𝟎

𝒉) 𝒍𝒐𝒈_𝟏𝟎𝒙 = 𝟑

4

WB5 investigate

Think about the function 𝑦 = 10^𝑥. Every time we increase by one, we multiply y by 10.

By how much do we multiply y when we increase x by 0.5 ?

Given that √𝟏𝟎 ≈ 𝟑. 𝟏𝟔 and √𝟏𝟎^𝟒

≈ 𝟏. 𝟕𝟖 complete the table below

10⁰= 10^0.25 ≈ 10^0.5≈ 10^0.75 ≈

10¹= 10^1.25≈ 10^1.5≈ 10^1.75≈

10²= 10^2.25 ≈ 10^2.5≈ 10^2.75 ≈

10³= 10^3.25 ≈ 10^3.5≈ 10^3.75 ≈

What does the whole number part of the power of ten tell us about the value of y ? How is this connected with standard form?

Can you find approximate solutions to the following equations:

1) 10

^𝑥

= 2

2) 10

^𝑥

= 5000

3) 10

^𝑥

= 562000

4) 10

^𝑥

= 0.00178

5

WB6 𝑙𝑜𝑔_an = x mean that 𝑛 = 𝑎^𝑥

Find the value of:

𝑎) 𝑙𝑜𝑔₃82

𝑏) 𝑙𝑜𝑔₄0.25

𝑐) 𝑙𝑜𝑔₁₀100

𝑑) 𝑙𝑜𝑔749

𝑒) 𝑙𝑜𝑔_0.54

𝑓) 𝑙𝑜𝑔_𝑎(𝑎⁵)

𝑔) 𝑙𝑜𝑔₃(3⁸)

ℎ) 𝑙𝑜𝑔_0.5(2³)

WB7

a) Write 𝑙𝑜𝑔8(64) = 2 in the form 𝑎^𝑏 = 𝑐 Hence find 𝑙𝑜𝑔8(64)

b) Write 𝑙𝑜𝑔₅(78125) = 5 in the form 𝑎^𝑏 = 𝑐 Hence find 𝑙𝑜𝑔₅(78125)

WB8 ‘a’ is known as the ‘base’ of the logarithm… 𝑙𝑜𝑔_an = x mean that 𝑛 = 𝑎^𝑥 Write these as logarithms

𝑎) 2⁵= 32

𝑏) 10³= 1000

𝑐) 5⁴ = 625

𝑑) 2¹⁰= 1024

𝑒) 5⁻²= ¹

25

𝑑) 6⁻³= ¹

216

6

WB9 Evaluate the following. Check with your calculator

WB10 Choose a number from the box for the base of each of these logarithms. Check with your calculator

a)

log

...

16 = 2

b) log

...

64 = 3 c) log

...

25 = 2

d) log

...

1 = 0

e) log

...

1/9 = -2

f) log

...

100 = 2

g) log

...

1/2 = -1

a). log

₄

64 b). log

₁₀

1000000

c). log

_1/2

1/8 d). log

_x

27 = 3

e). log

₁₀₀

1 f). log

₈

1/64

g). log

₂

2 h). log

₃

81

4 0

5 6

1 8

3 9

10 16

2

7

WB11

𝑙𝑜𝑔

₃

3 = 1 𝑙𝑜𝑔

₉

3 + 𝑙𝑜𝑔

₉

3 = 1 𝑙𝑜𝑔

₂₇

3 + ⋯ + 𝑙𝑜𝑔

₂₇

3 = 1

How many 𝑙𝑜𝑔

₈₁

3 do you need to add together to make one?

Can we choose integers 𝑥 and 𝑦 so that 𝑙𝑜𝑔

₆

𝑥 + 𝑙𝑜𝑔

₆

𝑦 = 1

How many different ways are there of doing this?

8

Laws of Logarithms

‘a’ is known as the ‘base’ of the logarithm 𝑙𝑜𝑔𝑎𝑛 = 𝑥 means that 𝑎^𝑥 = 𝑛

You do not need to know proofs of these rules, but you will need to learn and use them

WB12 Write each of these as a single logarithm:

𝑎) log₃6 + log₃7

𝑏) log₂15 − log₂3

𝑐) 2 log₅3 + 3 log₅2

𝑑) log₁₀3 − 4 log₁₀(1 2)

9

WB13 Write in terms of logax, logay and logaz

𝑎) log𝑎( 𝑥²𝑦𝑧³)

𝑏) log_𝑎(𝑥 𝑦³)

𝑐) log_𝑎(𝑥√𝑦 𝑧 )

𝑑) log_𝑎(𝑥 𝑎⁴)

WB14 Write terms of log a, log b and logc

𝑎) 𝑙𝑜𝑔 1 𝑎²

𝑏) 𝑙𝑜𝑔𝑎𝑏 𝑐

𝑐) 𝑙𝑜𝑔 √ 𝑎 𝑏𝑐²

𝑑) 𝑙𝑜𝑔 (10𝑎𝑏²)

10

WB15 Exam Q: Simplify

𝑎) 9 − 10 𝑙𝑜𝑔

_𝑎

𝑎

b) 𝑙𝑜𝑔

₂

𝑎

⁵

+ 𝑙𝑜𝑔

₂

√𝑎 𝑙𝑜𝑔

₂

𝑎

WB16 Exam Q: The point P on the curve

𝑦 = 3𝑘

^𝑥,

where k is a constant, has its y-coordinate equal to 9𝑘² Show that the x-coordinate of P may be written as

2 + 𝑙𝑜𝑔

_𝑘

3

11

Logarithms and Equations

When the power in an equation is an expression we ‘Take Logs’ on both sides of the equation. Then the rules of logarithms allow us to bring the expression to the front of the logarithm leading to a solution

WB17 Solve:

𝑎) 7

^𝑥

= 45

𝑏) 5

^𝑥

= 240625

𝑐) 8

^𝑥

= 3

𝑑) 20

^𝑥

= 720

12

WB18 Solve:

𝑎) 7

^𝑥+1

= 3

^𝑥+2

𝑏) 6

^𝑥

= 3

^𝑥+2

WB19 Solve:

𝑎) 3

^𝑥

= 10

𝑏) 5

^2𝑥

= 8

𝑐) 2

^−3𝑥

= 5

𝑑) 3

^𝑥+1

= 2

^𝑥−1

𝑒) 4(3

^𝑥

) = 120

𝑑) 5(2

^𝑥

) = 3(7

^𝑥

)

13

WB 20 Disguised Quadratics Solve:

𝑎) 5

^2𝑥

+ 7(5

^𝑥

) − 30 = 0

𝑏) 2

^2𝑥

− 7(2

^𝑥

) + 10 = 0

WB21 Find the values of x which satisfy each equation.

Give your answers to three significant figures.

𝑎) 5

^2𝑥

+ 24(5

^𝑥

) − 25 = 0

𝑏) 2

^2𝑥

− 4(2

^𝑥

) = 5

𝑐) 3

^2𝑥

= 5(3

^𝑥

) − 6

14

WB 22

Change of base Formula

You need to be able to change the base to solve some logarithmic equations. The specification says

‘Students may use the change of base formula’ to solve equations of the form

𝑎

^𝑥

= 𝑏

’

WB 23 Use the change of base formula to show that

if

2

^3𝑥−1

= 3

then

𝑥 =

log 3+log 2 log 2

15

WB24 Solve each equation using the change of base formula

𝑎) 𝑙𝑜𝑔

₅

𝑥 + 6𝑙𝑜𝑔

_𝑥

5 = 5

𝑏) 𝑙𝑜𝑔

₆

10 + 𝑙𝑜𝑔

₃

9 = 𝑥

𝑐) 𝑙𝑜𝑔

₃

𝑥 + 8𝑙𝑜𝑔

_𝑥

3 = 6

𝑑) 𝑙𝑜𝑔

₄

𝑥

²

+ 6𝑙𝑜𝑔

_𝑥

4 = 8

16

Modelling with logarithms

use the logarithm laws to show that if 𝑌 = 8𝑎

⁴

then log

_𝑎

𝑌 = log

_𝑎

8 + 4 if 𝑌 = 60(2

^𝑡

) then log

₂

𝑌 = log

₂

60 + t if 𝑌 = 𝐴(3

^𝑥/2

) then log

₃

𝑌 = log

₃

𝐴 +

^𝑥

2

17

WB25 A bacteria culture doubles every 15 minutes.

a) How long will it take a culture of 20 bacteria to grow to a population P of 163840 b) By taking logarithms find a linear equation for the bacterial growth in the form

𝑦 = 𝐴 + 𝐵𝑡

WB26 The graph represents the growth of a population of bacteria P, over t hours The graph has a gradient of 0.6 and y-intercept (0, 2)

A scientist suggests that this growth can be modelled by the equation

𝑃 = 𝑎𝑏

^𝑡 where a and b are constants to be found

a) Write the equation of the line b) Find the values of a and b c) Interpret the values of a and b

18

WB27 In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the

experiment were counted. N and T are expected to satisfy a relationship of the form

𝑵 = 𝒂𝑻

^𝒃, where a and b are constants

a) Show that this relationship can be expressed in the form

log

₁₀

𝑁 = 𝑚 log

₁₀

𝑇 + 𝑐

, giving m and c in terms of the constants a and/or b

b) The diagram shows the line of best fit for values of

log

₁₀

𝑁

plotted against values of

log

₁₀

𝑇

Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment

c) Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1 000 000.

d) With reference to the model, interpret the value of the constant a.

19

WB28 Investigation

The table gives the rank and population of five UK cities (London is rank 1)

The relationship between rank and population is modelled by the function 𝑹 = 𝑨𝑷^𝒏 where A and n are constants

a) Make a table and draw a graph of log R against log P showing the linear relationship between them (line of best fit)

b) Use your graph to estimate the values of A and n c)

City Birmingham Leeds Glasgow Sheffield Bradford

Rank (R) 2 3 4 5 6

Pop (P).

(2sf)

1000 000 730 000 620 000 530 000 480 000

20

WB29

The graph represents the moving average trend in value of Mr G’s stock in a financial services company (£s). T is in months after Jan 2018.

The graph has a gradient of -0.14 and goes through point

(2, 3.88)

This growth can be modelled by the equation

𝐶 = 𝑎𝑏

^𝑡 where a and b are constants to be found a) Write the equation of the line

b) Find the values of a and b c) Interpret the values of a and b

21

WB30

The Richter scale is used to represent the magnitudes of earthquakes. Since there are such large variations in magnitudes, the scale uses logarithms and is given as:

𝑀 = 𝑙𝑜𝑔

₁₀

(

^𝐼

𝐼𝑜

)

comes from

𝐼 = 𝐼

₀

× 10

^𝑀 M is the magnitude of the Earthquake

𝐼 is the Intensity of the earthquake

𝐼_𝑜 is the intensity of the smallest recordable earthquake (seismic wave amplitude 0.0001 mm)

a) An earthquake has Intensity such that 𝐼 is 1200 times 𝐼_𝑜

Find the magnitude M of the earthquake as given by the Richter scale?

a) In 1989 NEWCASTLE had an earthquake of magnitude 5.6

1989 SAN FRANSISCO also had an earthquake, it was of magnitude 7.1 Find the ratio ^𝐼𝑆𝐹

𝐼_𝑁 of the two earthquakes