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Objectives :

•To know what log means

•To learn the laws of logs

•To simplify logarithmic expressions

•To solve equations of the type ax=b

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We WILL meet the graph of y=ax and will see that it represents growth or decay.

Say possibly growth of Bacteria if x>0

Say possibly decay of radioactivity if x<0 So we will need to be able to solve

equations of the type b = ax

e.g. How would you solve

Ans: If we notice that 3101000

3 x

31010 x

We can use the same method to solve

813 x or

2552 x

122 xx4 x

433 x 22 55 x

then, (1) becomes

- - - - (1)100010 x

We need to write 75 as a power ( or index ) of 10.

Suppose we want to solve

7510 x

This index is called a logarithm ( or log ) and 10 is the base.

Our calculators give us the value of the logarithm of 75 with a base of 10.

87511010 xThe value is ( 3 d.p.) so,

8751

8751 x

Tip: It’s useful to notice that, since 75 lies between 10 and 100 ( or ), x lies between 1 and 2.

21 1010 and

The button is markedLog 75

Logarithm

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919683log3

logarithms

log264=6 because 26 = 64

log2(1/2)=-1 because 2-1 = ½

log21=0 because 20= 1

log2√2=1/2 because 21/2 = √2

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Logarithms to base 10

Any positive number can be written as a power of 10. Logarithms to base 10 are used in such subjects as:

 Chemistry pH value of a liquidPhysics power ratio – e.g. noise level –

decibel scale Earthquake measurement - Richter scale

(Logarithmic scales are also used for example to measure radioactive decay, pitch of musical notes, f-stops in photography, particle size in geology and population growth)

Log to base 10

Log10 10 = 1

Scientists only work with 2 specific bases (log, ln)We do not write the 10 as Log means to base 10

Log10=1 Log 0.1 = -1 Log1=0Log100=2 Log 0.01=-2Log1000=3 Log0.001=-3We can not take logs of negative numbers

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Log is an Inverse

Log is the inverse to 10x (last lesson)

(can show this when we learn the laws of logarithms)

Ln is the inverse of ex, (we will see this later)

e is a very important irrational number in maths and science, it has some very special properties!!

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So A logarithm is just an index

Solve the equation 10x = 4 giving the answer correct to 3 significant figures.

“x is the logarithm of 4 with a base of

10”

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logindex

Log 4 = 0.602 (3 sig fig) – on calculator

Laws of Logarithms

These are like the laws of indices (surprised NO!)

log a xy = log a x + log a y

log a x/y = log a x – log a y

log a x n = n log a x

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Some Important Rules

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Loga1=0

1log aa

xn

xx an

an

a log1

loglog 1

xxx aaa loglog1

log 1

These are the

Laws of Logs

a0=1

a1=a

Using these Rules- Simplify

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6log3log24log aaa

6log3log4log 2aaa

6log34log 2aa

6

34log

2

a

~ Loga6

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Express in terms of logax, logay, logaz

Loga = loga x3 – logay2z

Loga = loga x3 – (logay2 + logaz)

zy

x2

3

zy

x2

3

zy

x2

3

Loga = 3loga x – 2logay - logaz) zy

x2

3

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Do Ex 11A Page 325 q 1 - 5

We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs!

Solving ba x

52 xe.g.1 Solve

Solution: 52 x

2log

5log

10

10x

) s.f. ( 3322

5log2log 1010 x

We “take” logs

( Notice that 2 < x < 3 since ) 8242 32 and

5log2log 1010 x

We used logs with base 10 because the values are on the calculator. However, any base could be used. You could check the result using the “ln” button ( which uses a base you will meet in A2 ).

Using the “power to the front” law, we can simplify the l.h.s.

x)3(1001000 e.g.2 Solve the equation

Solution: We must change the equation into the form before we take logs.xab

Using the “power to the front” law:

x3log

10log

) s.f. ( 3102 x

x3log10log x)3(1001000 x310

Divide by 100:

Take logs:

3log10log x

Solving ba x

SUMMARY

bxba ax log

The Definition of a Logarithm

Solving the equation bna x

• “Take” logs

The “Power to the Front” law of logs:

xkx ak

a loglog

• Use the power to the front law

• Rearrange to find x.

• Divide by n

Exercises

143 x

14log3log 1010 x

( 2 d.p. )

1. Solve the following equations giving the answers correct to 2 d.p.(a) (b) 15122 x

4023log

14log

10

10 x

(a) “Take” logs:14log3log 1010 x

0898112log

15log2

10

10 x

(b) 15log12log 102

10 x“Take” logs:

15log12log2 1010 x

( 2 d.p. )540 x

2. Solve the equation giving the answer correct to 2 d.p.

x)2(200500

Solution: Divide by 200:xx 252)2(200500

x2log52log Take logs:

Power to the front: 2log52log x

Rearrange: x

2log

52log

( 2 d.p. )321 x

Exercises

Do Ex 11A page 326 no 6 ff

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