Consider the statement
1000 x 100 = 100 000
Remembering
(hopefully!) that….
103 = 1000
102 = 100
105 = 100 000
We can rewrite our original statement in
power (index) format as
103 x 102 = 105
Our statement that 102 x 103 = 105 is just a
specific case of the general rule
a mx a n = a m+n
Power Rule 1
In Year 10, you learned two other rules which
went hand-in-hand with 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
Rule 6………a -n =
=a
1/a n
Help! I’m
drowning in
rules!
Rule 7………a
1/n
n
= a
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….?
100 =
102
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….
SAME THING!
log10 100 = 2
So we now have these two
interchangeable formats……
100 = 102
This is
called
POWER
FORMAT
log10100 = 2
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 n
Replace 10 with a
Replace 2 with t
n = at
loga n = t
LEARN!
Log Law #1 – the most important of all!
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!).
Use the Transformation Rule to fill in this table
Power Format
9 = 32
81 = 34
64 = 43
216 = 63
2-2 = ¼
1/125 = 5-3
81/3 = 2
Click to check your answers
Log Format
log39 = 2
log381 = 4
log464 = 3
log6216 = 3
log2(1/4) = -2
-3 = log5(1/125)
log82 = 1/3
Find the value of log41024
Solution
Let log4 1024 = x
By the Transformation Rule logan =t
We can write
log4 1024 =x
n = at
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 = 5
Now
solve
this!
There are more technically correct ways to do this, but for the moment,
trial and error will do!
Evaluate without a calculator log
2
0.25
Solution
Let log 2 0.25 = x
By the Transformation Rule logan =t
We can write
log2 0.25 =x
n = at
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
Remember our original
statement on Slide 2 ?
1000 x 100 = 100 000
Which we then rewrote as
103 x 102 = 105
We can now go one better, and realising that the
powers can be connected using 3 + 2 = 5, we can
now write this another way, i.e.
log101000 + log10100 = log10100 000
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)
which can also be written as
log n a - log n b = log n (a / b)
Log Law
#3
Simplify log35 + log34
Solution
Using Log Law #2 i.e.
log n a + log n b = log n (ab )
Let n = 3, a = 5 and b = 4. This means ab = 20
So log35 + log34 = log320
(ans)
Note: This is only possible when the base (n) is the
same in both terms.
Simplify log272 – log29
Solution
Using Log Law #3 i.e.
log n a - log n b = log n (a / b )
Let n = 2, a = 72 and b = 9. This means a / b = 8
So 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=2x
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
Now for the last of the
“Big Four”
YAY!!
Remember Log Law #2 back on Slide 15?
log
n
(ab) = log
a + log
n
b
n
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
i.e.
n
(a
3
log
) = log
n
n
a + log
n
a + log
(a 3) = 3 log
n
a
n
a
Check that you
understand this
before next slide!
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 Law
log 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!
Again applying the Transformation
Rule gives 1 = a 0 which is true!
Law #6:
log
a
1 = 0
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
Solution
First 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 64
As 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
Solution
Again 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 5 225
As 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 !
4 log10 100
5log10 1000
Simplify
Solution
Remember 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.
4 log10 (10 2 )
4 log10 100

5log10 (10 3 )
5log10 1000
4  2log10 10

5  3log10 10
4  2(1)

5  3(1)
8

15
Now use Law #4 to drop
powers (2 & 3) to the
front
Now use Law #5
Ans!
If log4
1
3 3
 a log4 3, find the value of a
Solution
First use Law 3 to change the left side and kill the
fraction
log4
1
3 3
 log4 1  log4 (3 3) Now use Law #6 loga1
 log4 (3 3)
3
  log4 (3 2 )
= 0
Now remember 33 =
3 x 31/2 = 3 3/2
Now Law#4 to bring power
down front
3
  log4 3
2
which is now of form a log4 3 so a = -3/2
Express in simplest form 4 – 3log10 x
Solution
This 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…
4 – 3 log
10
x
Hmmmmmm….
what to do
with the 4 ???
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
log
10 y = 4
n = a
t
y = 10
4
This means that y = 10 000 and so 4 = log10 10000
So 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
Solution
Remember 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
Use Law #4 to move the 3 to front
= 5 x 3 log 2 2 + 3
Use Law #5 to simplify log22
= 5 x 3 (1) + 3
=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
Use LOG LAW #1: Transformation Rule
log
a
n = t
n = a
t
log2 x = 5
so x = 25
x = 32
Easy!!
Solve log
3
(1/9) = x
Solution
Use LOG LAW #1: Transformation Rule
log
a
n = t
n = a
t
log3 (1/9) = x
so 1/9 = 3x
i.e. 3-2 = 3x
Equating the powers,
x = -2
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 sides
Now use Law#4 on left expression
x log10 3 = log1020 Finally divide both sides by log10 3
to make x the subject
log10 20
x 
log10 3