Consider the statement

1000 x 100 = 100 000

We can rewrite our original statement in power (index) format as

103 x 102 = 105

Remembering (hopefully!) that….

103 = 1000

102 = 100

105 = 100 000

Our statement that 102 x 103 = 105 is just a specific case of the general rule

a m x a n = a m + n

In Year 10, you learned two other rules which went hand-in-hand with Rule 1….

Power Rule 1

a m ÷ a n = a m – n Power Rule 2

(a m )n = a m n Power Rule 3

These last two rules can be easily verified using real numbers as we did on the previous slide. These three rules are the basis of all our logarithm work to come. MAKE SURE YOU KNOW THEM!!

There are also some other rules you need to remember as these appear in log work….Rule 4………a 0 = 1

Rule 5………a 1 = a

Rule 6………a -n = 1/a n

Rule 7………a 1/n = a

n

Help! I’m drowning in

rules!

Now let’s take this a bit further and go beyond Year 10 work…..

In the statement

100 = 102

100 is called the “number”

10 is called the “base”, and

2 is called the “logarithm” ( known to Year 10s as “power” or “index” but in senior school we call it LOGARITHM !!

So LOGARITHM is just a fancy word for POWER

So, where are we….?

can be described in words as “100 equals 10 squared”.

BUT…using our new terminology from the earlier slide we can also say

2 is the LOGARITHM of the NUMBER 100 (BASE 10)

and in symbols….

100 = 102

log10 100 = 2SAME THING!

100 = 102 log10100 = 2

So we now have these two interchangeable formats……

This is called

POWER FORMAT

This is called

LOGARITHMIC FORMAT

Now, if we replace the 100, 10 and 2 with letters, we can come up with a formula which then enables us to do this interchange for all numbers, bases and logarithms

100 = 102 log10100 = 2

Replace 100 with nReplace 10 with a

Replace 2 with t

n = at loga n = t

This is known as the TRANSFORMATION RULE and must be memorised! It will enable you to

swap between power format and log format with ease!Insect lovers take note! You might notice two insects, ANTs (which

live in logs) and NATs (misspelt!).

LEARN!

Log Law #1 – the most important of all!

Use the Transformation Rule to fill in this table

Power Format Log Format

9 = 32

log381 = 4

64 = 43

log6216 = 3

2-2 = ¼

-3 = log5(1/125)

81/3 = 2 log82 = 1/3

log39 = 2

log464 = 3

log2(1/4) = -2

81 = 34

1/125 = 5-3

216 = 63

Click to check your answers

Find the value of log41024

Solution

Let log4 1024 = x

By the Transformation Rule logan =t n = at

We can write log4 1024 =x 1024 = 4x

It’s easier to solve power format than log format. Using calculator we trial various powers of 4, and ultimately we find that 45

= 1024, so x = 5There are more technically correct ways to do this, but for the moment, trial and error will do!

Now solve this!

Evaluate without a calculator log 2

0.25Solution

Let log 2 0.25 = x

By the Transformation Rule logan =t n = at

We can write log2 0.25 =x 0.25 = 2x

Now if we change 0.25 into ¼, which is 1/22 or 2-2, we then see that 2x = 2-2, and so x = – 2

1000 x 100 = 100 000

Remember our original statement on Slide 2 ?

103 x 102 = 105

Which we then rewrote as

We can now go one better, and realising that the powers can be connected using 3 + 2 = 5,

log101000 + log10100 = log10100 000

we can now write this another way, i.e.

log101000 + log10100 = log10100 000

Now if we replace

1000 with a

100 with b

100 000 with a x b

10 with n

We can now write a general formula……

log n a + log n b = log n (a x b)

Log Law #2

This is really just a disguised version of the Year 10 rule that when you multiply, you add the powers!

Using a similar approach we can also show that

log n a - log n b = log n (a ÷ b) Log

Law #3which can also be written as

log n a - log n b = log n (a / b)

Simplify log35 + log34

Solution

log n a + log n b = log n (ab )

Using Log Law #2 i.e.

Let n = 3, a = 5 and b = 4. This means ab = 20So log35 + log34 = log320 (ans)Note: This is only possible when the base (n) is the same in both terms.

Simplify log272 – log29

Solution

log n a - log n b = log n (a / b )

Using Log Law #3 i.e.

Let n = 2, a = 72 and b = 9. This means a / b = 8So log272 – log29 = log28 (ans – almost!)Now you should always check to see if these two numbers (the 2 and 8) are related in any way…… see next slide!

Our answer is log28 but it can be simplified !

If you can get into the habit of checking if the 2 (base) and the 8 (number) are related as powers, you are then able to use the Transformation Rule….

log28 = x 8 = 2 x

And so x = 3

A better ans then is:

log272 – log29 = log28 =

3

Note we could not have done this in Example 3 as 20 and 3 are not related.

Now for the last of the “Big Four”

Remember Log Law #2 back on Slide 15?

log n (ab) = log n a + log n b

This can be extended to more than two terms, e.g.

log n (abc) = log n a + log n b + log n c(3 terms)

Or if a, b, c are all the same, say they’re all “ a ”…..then

log n (a 3 ) = log n a + log n a + log n a

i.e. log n (a 3) = 3 log n a

Check that you understand this before next slide!

YAY!!

And if there are 4 terms, then….

log n (a 4) = 4 log n a

And if there are “ y ” terms we can generalise to get our Log Law #4 Formula….

log n (a y ) = y log n a

Log Law #4

LOG LAW #1: Transformation Rule

log a n = t n = a t

LOG LAW #2: When numbers are multiplied, you ADD their logs

log a (xy) = log a x + log a y

LOG LAW #3: When numbers are divided, you SUBTRACT their logs

log a (x / y) = log a x - log a y

LOG LAW #4: The Power Lawlog a (x n ) = n log a x

to the “Big 4”, there are also some “lesser” log laws which are special cases of the Big 4 and come in extremely handy!

Law #5: log a a = 1

If you apply Law #1 (the Transformation Rule), you will see that a = a 1 which is certainly true!

Law #6: log a 1 = 0

Again applying the Transformation Rule gives 1 = a 0 which is true!

Law #7: log a (1/x) = - log a

x Using Law #3 we first get log a 1 – log a x and then using Law #6 above this becomes 0 – log a x i.e. – log a x

Now we’ll do some examples which require all

7 log laws to be used strategically!

Simplify 2log35 + 3log34

SolutionFirst use Law #4 to shift the coefficients (2 & 3) up to the power position

2log35 + 3log34

= log3 (52) + log 3 (43) which is log 3 25 + log 3 64As this is now of the format log a + log b we can use Law #1 to combine together and get log (ab)

= log3 (25 x 64)Work it out

= log3 (1600)

At this stage, check if there is any recognisable power connection between 3 and 1600. Maybe check powers of 3 on the calc. There appears to be no connection, so leave this as the ans.

Simplify 2 log 5 3 - 2log 5 15

SolutionAgain use Law #4 to shift the coefficients (2 & 2) up to the power position

2log 5 3 - 2log 5 15

= log5 (32) - log 5 (152) which is log 5 9 - log 5 225As this is now of the format log a - log b we can use Law #2 to combine together and get log (a / b)

= log5 (9 / 225)Work it out

= log5 (1/25)

At this stage, check if there is any recognisable power connection between 5 and 1/25. Now the 5 and 25 should give you the clue: 1/25 is equal to 5-

2 !! OVER…

To work out log5 (1/25)

Rewrite as

log5 (5 -2 )

Use Law #4 to drop the power down the front

= -2 log5 5

Now use Law #5 log a a = 1 which can only be used then the base and number are the same!! (here they’re both 5!)= -2 x 1

= -2 Ans !

Simplify 1000log5100log4

10

10

SolutionRemember if you can spot a connection between the base (10) & numbers (100 & 1000), always work on this first, so rewrite the numbers as powers of 10.

1000log5100log4

10

10

)10(log5)10(log4

310

210

Now use Law #4 to drop powers (2 & 3) to the front

10log3510log24

10

10

Now use Law #5

)1(35)1(24

158

Ans!

Solution

First use Law 3 to change the left side and kill the fraction

)33(log1log33

1log 444 Now use Law #6 loga1 = 0

If 3log33

1log 44 a , find the value of a

)33(log4 Now remember 33 = 3 x 31/2 = 3 3/2

)3(log 23

4Now Law#4 to bring power down front

3log23

4

which is now of form a log4 3 so a = -3/2

Express in simplest form 4 – 3log10 x

SolutionThis is the style of Q14, P283. The question is asking you to write 4 – 3 log 10 x as a single log, i.e. in format log 10 a.

The overall strategy is firstly to write the “4” as log 10 (something) and use Law #4 to move the 3 to the power position. This will then give us a format log a – log b which we can then switch to log (a/b) using Law #3. PHEW!!! Here we go…

Hmmmmmm…. what to do with the

4 ???

4 – 3 log 10 x

Since there’s already a “log 10” present, maybe we could write the 4 as log 10 (something) ??

So Let 4 = log 10 y

Applying Law #1 (Transformation Rule)

log a n = t n = a t

log 10 y = 4 y = 10 4

This means that y = 10 000 and so 4 = log10

10000So back to the original question

4 – 3 log 10 x can now be rewritten as

log10 10000 – log 10 x 3

using Law #4 to move the 3= log10 (10000 /x

3)using Law #2

Simplify 5log28 + 3

SolutionRemember to first look for a connection between the 2 and 8 ? As 8 = 23 we can write log 2 8 = log 2 (23)

5log28 + 3

= 5log2 (23) + 3

= 5 x 3 log 2 2 + 3

Use Law #4 to move the 3 to front

= 5 x 3 (1) + 3

Use Law #5 to simplify log22

=18 NOTE!! Here we didn’t have to change the “3” on the end into log (something), as we were able to simplify the first term and get rid of the log. This was possible because we made the effort to first find that connection between the 2 and the 8!

First revise:

Transformation Rule

Negative Indices

Fractional Indices

Solve log 2 x = 5

Solution

log2 x = 5

so x = 25

x = 32

Use LOG LAW #1: Transformation Rulelog a n = t n

= a t

Easy!!

Solve log 3 (1/9) = x

Solution

log3 (1/9) = x

so 1/9 = 3x

x = -2

Use LOG LAW #1: Transformation Rulelog a n = t n

= a t

i.e. 3-2 = 3x

Equating the powers,

This is why I asked you to revise negative powers!!

Solve 3 x = 20

Note: This is a very common question where the unknown is in the power and there is no obvious connection between the two numbers (3 and 20 in this case). The strategy is to take log10 of BOTH SIDES then use LAW #4.

3 x = 20

log10 (3x) = log1020

First, take logs10 of both sidesNow use Law#4 on left expressionx log10 3 = log1020Finally divide both sides by log10 3 to make x the subject

3log20log

10

10x