Add Maths Logs Final

8/3/2019 Add Maths Logs Final

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Logarithms

A Logarithm is just another way of writing an index

or power!

8log 3 is read as log of 8 to base 3

6log10 is read as log of 6 to base 10. If the base is

not written, it can be assumed that it is base 10. e.g.

4log is read as log of 4 to base 10. If it is any otherbase it must be written in.

It is important to remember the following law:

e.g. 10010..2100log 210 == ei1000010..410000log

410 == ei

532log322 25 == then

Example 1

Express the following in index form:

(a) 29log 3 = (b) rqp =log

NB: If no base is written in, assume base 10.

If xna =log then nax =


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Example 2

Rewrite as a logarithm or in index form:

(a) 1000103 =

(b) 1024210 =

The properties of logarithms

(1) yxxy aaa logloglog +=

(2)yx

y

xaaa logloglog =

(3) xnx ana loglog =

(4) 01log =a

(5) 1log =aa

Example 3

Evaluate:

(a) log63 (b) log1000 (c) log3.7

Example 4

Solve for x

(a) Log x = 2.1 (b) log x = -1.7


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Example 3

Evaluate without using a calculator

(1) log10000

(2) log 310(3) log 100 10

(4) log 100

310

Example 4

Find the value of :

1.) =81log 3 2.) =25.0log 4

3.) =4log 5.0


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Example 5

Express the following in terms of loga, log b and log c

(1) log(abc)

(2) log(a/bc)(3) log(abc)

(4) log(a/bc)

(5) log(10a2bc3)


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Example 6

Given log2 =0.3010 and log 3 = 0.4771, find, without the

use of the log function on your calculator, the value of

(a) log8(b) log(30)1/3

(c) log24

PPQ 2009

(a) Write 3 log p + 4 log q as a sing logarithm.

(b) Write log c

d

PPQ 2007

(b) If log 6 = p and log 4 = q express log 36 in terms of

p and hence log 9 in terms of p and q. (3)


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PPQ

(a) If =81log 3 a what is the value of a?

(b) If =2log 3 b express 18log 3 in terms of b?

PPQ 2010

(c) If =5log 3 and =8log 3 express the following in terms

of a and b.

(1) =6.1log 3

(2) =120log 3

Solving Equations using logarithms


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These equations are normally solved by taking logs of

both sides, using are knowledge of algebra and the

rules of logs.

Examples

Solve for x

(1) 2x = 9

(2) 83x

= 256

(3) 32x + 3 = 76


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(4)4x 1 = 3x 2

(5) 55x-1 = 31-2x


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PPQ 2002

Solve the equation3(2x - 1) = 8

Giving your answer correct to 3 decimal places (2

marks, now you get 4 marks for solving something like

this)

PPQ 2003Solve the equation

0.1(1 - x) = 7

Giving your answer correct to 3 decimal places (2

marks, now you get 4 marks for solving something like

this)

Reduction of a law of the form P = kln to linear

form using logarithms


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Suppose a table of values of quantity P is available at

various values of quantity l. It is thought that P is

related to l via the law P = kln, where k and n are to bedetermined. How can it be confirmed that the law

holds and what are the values of k and n?

Taking Logs of both sides gives us

Log P = Log (kln)

And using are rules of Logs gives us

Log P = Log k + Log ln

Log P = Log k + n Log l

If the table of values is reconstructed in terms of LogP and Log l and the tabulated pairs plotted, the

straight line Y = mX + c results

Log P = n Log l + Log k

Y = m X + c

So if we plot Log P against Log l, we should get a

straight line, where m (which is the gradient) will be

are n value and c which is the intercept on the y-axis

will be are Log k value, hence we will be able to find k.

PPQ 2002


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A small landing aircraft leaves a main space station

orbiting Mars and descends to the surface of the

planet. The table below shows the altitude ametres at

which at which the craft should be when it is travellingat velocity vm/s.

Velocity v (m/s) Altitude a (m)

3500 154060

2500 129770

2000 115810

1500 1000001000 81320

It is believed that a relationship of the form

a = kvn

exists between aand v, where k and n are constants.

(a) Verify this relationship by drawing a suitable

straight line graph, using values correct to 3

decimal places. Label the axes clearly(6)(b) Hence, or otherwise, obtain values for k and n.

(4)

(c) Using the formula a = kvn with your values for k

and n, calculate the altitude of the craft when

its velocity is 2800 m/s. (2)

(d) The craft is due to reach subsonic speed at an

altitude of 39000m. Calculate the velocity ofthe craft when it is at this height. State the

assumption which you have made.(3)

PPQ 2003


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An ornithologist measured the weights of birds eggs

and compared these with the body weights of adult

birds. The table below shows the body weight bgrams

and the egg weight egrams for several birds.Bird Bird weight b

(grams)

Egg weight e

(grams)

Sparrow 30 18.4

Thrush 70 26.9

Tern 130 35.6

Puffin 400 59.0Heron 1600 110.1

Stork 3400 154.5


e = kbn

exists between eand b, where k and n are constants.

(a) Verify this relationship by drawing a suitable

straight line graph, using values correct to 3decimal places. Label the axes clearly(6)

(b) Hence, or otherwise, obtain values for k and n.

(4)

(c) Using the formula e = kbn with your values for k

and n, calculate the egg weight for an

oystercatcher, whose body weight is 600 grams

(2)

(d) Using the formula e = kbn with your values for k

and n, calculate the body weight of an owl whose

egg weight is 53.4 grams. (3)

PPQ 2008


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Cepheid variables are stars whose brightness

fluctuates regularly with time. An astronomer

measured the period P and the average luminosity

(intrinsic brightness relative to the sun) L of severalstars and the results are given in the table

Period P (days) Average luminosity L

(suns)

5.32 2017

11.61 4947

25.18 12050

38.49 19630

51.42 27390


L = kPn

Exists between L and P, where k and n are constants.

(1) Verify this relationship by drawing a suitable

straight line graph, using values correct to two

decimal places. Label the axes clearly. (6)

(2) Hence, or otherwise, obtain values for k and n.

(4)

(3) Use the formula L = kPn with the values youobtained for k and n to calculate the average

luminosity of the star X-Cygni, which has a

period of 16.39 days. (1)

(4) Use the formula L = kPn to calculate the period

of the star Y-Cygni, whose average luminosity is

1394 suns. State any assumptions about theformula which you make. (3)


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Add Maths Logs Final

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