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January 2001

Number 10

Exponentials and Logarithms Fig 2. Graphs of y = 2x, y = 4x and their gradients

Exponentials and logarithms are used in a number of areas of Physics, including radioactive decay and capacitor charge and discharge.

gradient of y = 4x

This Factsheet will explain what exponentials and logarithms are, the rules for their manipulation and how to use these functions on a calculator. It will also describe the use of logarithmic scales for graphs. y=2x

Before starting on this Factsheet, you should ensure you are familiar with the rules for manipulating indices, covered in the Factsheet 8 – Indices, Standard Form and Orders of Magnitude.

y = 4x

gradient of y=2x

1. The Exponential Function and its graph Exponentials are related to powers (or indices). To get an idea what is involved, we will look at the graph of y = 2x. We can plot some points on this graph by putting in values of x, as in the table below: x y=2x

-3 0.125

-2 0.25

-1 0.5

0 1

1 2

2 4

3 8

You will notice from fig 2 that the gradient of y = 2x is always below the original curve, and the gradient of y = 4x is always above the original curve, but that the gradient curves are the same sort of shape as the originals.

4 16

This suggests that there is perhaps a number between 2 and 4 for which the gradient curve and the original curve are exactly the same. It turns out that there is such a number – it is called e and is equal to 2.71828....... Like , e goes on for ever without repeating.

This gives us the graph shown in fig 1. Fig 1. Graph of y = 2x 15 15

The graph of y = ex is very similar to that of y = 2x or y = 4x (fig 3). ex is called the exponential function

10 10

Fig 3. Graph and key properties of the exponential function

55

y = ex -2 -2

0

0

0

2

42

4

We can see some important properties of 2x from this graph:
It is always positive
For negative values of x, it approaches 0
As x gets larger, 2x gets large very rapidly.

(0,1)

You could similarly plot graphs of y = 3x, y = 10x etc. All of these will have the same 3 properties listed above.

(1, 2.71828....)

Key properties of ex

For each of these graphs, you could find the gradient at any particular point by drawing a tangent. You could then draw up a table showing the gradient at a number of different x-values, and plot a graph of this. Fig 2 shows the graphs y = 2x and its gradient, and the graph y = 4x and its gradient.







1

It is always positive As x becomes more
ve, ex approaches, but doesn’t reach, 0 As x becomes large and +ve, ex gets large very quickly (more quickly than any power of x) The gradient of y = ex at any point is equal to the y-value at that point


   

Exponentials and Logarithms  In Physics, you will most often meet a negative exponential graph. These are the same shape as y = e-x (this is not the same as y = -ex). This looks like the normal exponential graph reflected in the y-axis

2. Logarithms Logarithms are really another way of writing powers. For example, we know 102 = 100. We write this using logarithms as log10 100 = 2, which reads “log to the base 10 of 100 is 2”

In applications such as radioactive decay or capacitor discharge, the graph you need will actually be of the form y = Ae-bt, where A and b are numbers, and t is time – so something like y = 8e-0.6t. Also, you will usually only need the part of it for which t is positive. Fig 4 gives some key properties of such graphs.

Similarly, if we write log4 64 = 3, this is another way of writing 43 = 64 So generally:

Fig 4. Negative exponential graphs and their properties y

loga x = b

 x = ab

(0, A) Manipulating logarithms There are three laws of logarithms – they are related to the laws of indices: y = Ae-bt


t

Key properties of negative exponential graph of the form y = Ae-bt
The “initial” value of y – where the curve crosses the y-axis – is A


As t increases, y decreases rapidly towards 0, but never reaches 0.




Any process modelled by such a graph has a constant “half-life” – it always takes the same time for the value of y to halve. So, for example, if y was initially equal to 8, and 5 seconds later was equal to 4, then in another 5 seconds y will decrease to 2.

ii) e-2x  e6x

a
=loga – logb b




log 




(if you are dividing the numbers, subtract the logs) log(an) = nloga (you can “bring down the power”)

In addition, you need to know the following key facts about logarithms:






Manipulating exponentials Since exponentials are a special case of powers, you use the same rules as for manipulating powers. Don’t let the “e” scare you! Example 1. Simplify: i) (2ex)2

log(ab) = log a + log b (so if you are multiplying the numbers, you add the logs)




iii) e4x  e-3x

You cannot have the log of a negative number or zero loga a = 1 – so log10 10 = 1. (this is because 10 = 101) log 1 = 0, for any base of the logarithm (this is because any number to the power 0 is 1) You cannot simplify log(x + y) or log(x – y) or anything else with a + or – sign in the bracket.

i) (2ex)2 = 22 (ex)2 = 4e2x ii) e-2x  e6x = e-2x + 6x = e4x iii) e4x  e-3x = e4x
-3x =e7x

As with indices, most problems arise from students trying to invent their own rules! Stick to the ones above – if they don’t tell you how to simplify it, then you probably can’t!

Exponentials on your calculator Your calculator will have a key marked ex (you may need to get at this by doing 2nd or INV and then lnx). NB This is NOT the key marked EXP!

Example 2: Express in terms of simpler logarithms

 2x 

y 

i) log(x2)

ii) log(xy)

iii) log

iv) log(3x4)

v) log(Axn)

vi) logaa8

2

It depends on the calculator exactly how this works – to find e , you will have to do one of:
2 ex
ex 2 Check which it is on your calculator now! You should get 7.389...

i) Since a power is involved, we use the third log law: log(x2) = 2logx st ii)1 law: log(xy) = logx + logy  2x 
= log(2x) – logy
 y 

iii) 2nd law: log 

Calculators treat exponentials like indices in terms of order of operations. So if you do ex 2  3, it will find e2, then multiply it by 3.

= log2 + logx – log y (by 1st law) iv) 1st law: log(3x4) = log3 + logx4 = log3 + 4logx, by 3rd law v) 1st law: log(Axn) = logA + logxn = logA + nlogx, by 3rd law vi) 3rd law: logaa8 = 8logaa = 8  1 = 8.

If you are trying to find the exponential of a negative number, be careful when you put in the negative sign – you should always get a positive answer. (Try working out e-2 = 0.135...)

Tip: Many students are tempted to say log(3x4) = 4log(3x). This doesn’t work because only the x is to the power 4, not the 3.

2


   

Exponentials and Logarithms  Example 3. Express as a single logarithm i) log2 + log5 + log 3 ii) log 20 – log 5

Tip. If you are dealing with this sort of equation and you end up trying to simplify ln(-3ex), or anything else with a number or minus sign in front of the e – STOP! Rearrange the equation so the esomething is completely on its own first. You are much less likely to make mistakes this way!

iii) 2log3 + log8 – log12

i)1st law: log2 + log5 + log3 = log(2  5  3) = log30 ii) 2nd law: log20 – log5 = log(20  5) = log4 iii)3rd law: 2log3 + log8 – log12 = log32 + log8 – log12 = log(32  8) – log12 = log72 – log12 = log(72  12) = log 6

Finding logs on your calculator Calculators have buttons for log10 (written log) and natural logarithms (ln). Do not get the two confused! Calculators vary – with some you have to press the log or ln button then the number, and with others it’s the other way round. Check which yours is – try to find ln2 (= 0.693....)

The logarithms that are actually used are to the base 10 (written log or lg) to the base e (written ln, which stands for natural logarithm).

Calculators treat logs with indices in the order of operations. If you type in ln4  2, it will work out ln4, then multiply the answer by 2.

In Physics, you will mainly be concerned with natural logarithms, since they are helpful when you are dealing with equations that have e in them.

3. Logarithmic scales Sometimes graphs are required for data that cover a very wide range of values – going, for example, from 0.1 to 10000. One example of this is exponential decay This presents a problem, since if the largest values are to fit on the graph, it will be very hard to plot the small ones accurately.

Natural logarithms (lnx) Fig 5. shows the graph of y = lnx and some of its key properties. You may notice that the graph looks like the graph of y = ex, but reflected in the line y = x.

Logarithmic scales are used to overcome this problem. Instead of the actual value, the logarithm of the actual value is plotted. This can be done on one or both axes. Logarithms to base 10 are usually used for this purpose.

Fig 5. Graph and key properties of y = lnx y

y = lnx (e, 1)

This helps because the logarithms will not vary so much in size as the actual data values – for example, log 0.1 = -1 and log10000 = 4, so it would be easy to fit both of these on one graph.

x (1, 0)

Example 5. Plot a suitable logarithmic graph of the following data x y

1 0.006

2 0.5

3 1

4 3

5 25

6 120

7 900

8 4500

9 10002

10 99870

Only the y-values vary widely, so we only use a logarithmic scale on the y axis. First we find the values of logy:

Key properties of lnx









x y logy

ln1 = 0 lne = 1 lnex = x for any value of x lnx is not defined for x 0 lnx is negative for x < 1 lnx increases as x gets larger, but only slowly

1 0.006 -2.2

2 0.5 -0.3

3 1 0

4 3 0.48

5 25 1.4

6 120 2.1

7 900 3.0

8 4500 3.7

9 10002 4.0

10 99870 5.0

Then we plot the values of logy against x. However, on the scale, we mark the original values of y as well as (or instead of) the values of logy. This is easy to do when we are using log to base 10, since if logy = 5, y = 105 etcetera: Fig 6. Logarithmic scaled graph

We can use natural logarithms to solve equations with an unknown in the power: (e.g. exponential decay)

y lny 106 6

Example 4. Find the value of t for which 3 = 5e-2t

104

4

102

2

Step 2. Take the natural logarithm (ln) of both sides of the equation – do not try to simplify at this stage ln0.6 = ln(e-2t)

100

0

Step 3. Use laws of logs to simplify ln0.6 = -2t (by using lnex = x)

10-4 -4

Step 1. Rearrange the equation so the esomething part is on its own Dividing by 5, we get 0.6 = e-2t

10-2 -2

Step 4. Use your calculator to find t t = ln(0.6)  -2 = 0.255 (3SF)

3

x 0

2

4

6

8

10


   

Exponentials and Logarithms  You may also be given logarithmic graph paper e.g. when plotting intensity (lux) against LDR resistance ()This can either be logarithmic on one axis (which we would have used above) or on both axes). Fig 7 shows log-log paper

Questions 1. Sketch the following graphs, showing the coordinates of any points at which they cross the coordinate axes: b) y = lnx c) y = 4e-2x a) y = ex

Fig 7 Log-log graph paper 2. Simplify the following a) e4x  e –6x b) e7x  e1.5x

1000

c) (e4x)½

d) (3e2x)3

3. a) Given that a = b8, what is logba ? b) Express the following in terms of simpler logarithms: 

i) log(ab2)

100

ii) log 

z

iv) log10(100x2)

2




v) log1

iii) ln(5e2x) vi) lne4x

c) Express each of the following as a single logarithm i) log6 – log2 + log5 ii) 6log2 – 2log4 + log ½ iii) –2logx iv) 1 + ln2

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4. Obtain the solution to each of the following: b) 0.8 = e-3x c) 100 = 15e0.5x a) 2 = 3e4x

1 1

10

100

1000

Answers 1. a), b) see graphs in the text. c) -ve exponential graph, intercept (0, 4) c) e2x d) 27e6x 2. a) e-2x b) e5.5x 3. a) 8 b) i) loga + 2logb ii) logx + logy – 2logz iii) ln5 + 2x v) 0 vi) 4x iv) 2+ 2log10x c) i) log15 ii) log2 iii) logx-2 iv) ln(2e) 4. a) –0.101 b) 0.0744 c) 3.79 d) 0.304 (all to 3 SF) 5. Either (on log-normal paper):

In the figure, the scales shown go from 1 to 1000. You could change this, but you must always have the powers of 10 (1000, 100, 10, 1, 0.1 etc) in the same places on the graph paper – so, for example, the scale on the x-axis could read 0.1, 1, 10, 100 instead. Example 6. Use log-log graph paper to plot a graph of the following data 0.09 9.8

0.3 86

3 520

12 1600

48 4600

d) 6e-0.6x – 5 = 0

5. Plot a suitable logarithmic graph for the following data. x 200 1 500 10 800 99 700 800 000 3 460 000 y 4 7 9 12 16 18

When you use logarithmic graph paper, you plot the actual x and y-values, but because the divisions in the paper are not of equal size, the scale is actually logarithmic.

x y

xy 

20

98 9800

15 10 5

We label the x-axis with 0.01, 0.1, 1, 10, 100 and the y-axis with 1, 10, 100 and 1000. The divisions between 0.01 and 0.1, for example, correspond to 0.02,0.03.... 0.09, so we plot the points accordingly

0 1

100

10000

1000000

100000000

or (on normal graph paper)

Fig 8. Log-log graph y 20

10000 15 10

1000 5 0

100

00 10

22 10

1044

66 10

1088

x

10

1 0.01

0.1

1

10

Acknowledgements: This Factsheet was researched and written by Cath Brown Curriculum Press, Unit 305B The Big Peg, 120 Vyse Street, Birmingham B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. They may be networked for use within the school. No part of these Factsheets may be reproduced, stored in a retrieval system or transmitted in any other form or by any other means without the prior permission of the publisher. ISSN 1351-5136

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