Consider the statement 1000 x 100 = We can rewrite our original statement in power (index) format as 10 3 x 10 2 = 10 5 Remembering (hopefully!) that… = = =

Our statement that 10 2 x 10 3 = 10 5 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 = 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 = 10 2 log = 2 SAME THING!

100 = 10 2 log = 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 = 10 2 log = 2 Replace 100 with n Replace 10 with a Replace 2 with t n = a t log a 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 FormatLog Format 9 = 3 2 log 3 81 = 4 64 = 4 3 log = = ¼ -3 = log 5 (1/125) 8 1/3 = 2 log 8 2 = 1/3 log 3 9 = 2 log 4 64 = 3 log 2 (1/4) = = 3 4 1/125 = = 6 3 Click to check your answers

Find the value of log Solution Let log = x By the Transformation Rule log a n =t n = a t We can write log =x 1024 = 4 x It’s easier to solve power format than log format. Using calculator we trial various powers of 4, and ultimately we find that 4 5 = 1024, so x = 5 There 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 Solution Let log = x By the Transformation Rule log a n =t n = a t We can write log =x 0.25 = 2 x Now if we change 0.25 into ¼, which is 1/2 2 or 2 -2, we then see that 2 x = 2 -2, and so x = – 2

1000 x 100 = Remember our original statement on Slide 2 ? 10 3 x 10 2 = 10 5 Which we then rewrote as We can now go one better, and realising that the powers can be connected using = 5, log log = log we can now write this another way, i.e.

log log = log Now if we replace 1000 with a 100 with b 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 #3 which can also be written as log n a - log n b = log n (a / b)

Simplify log log 3 4 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 = 20 So log log 3 4 = log 3 20 (ans) Note: This is only possible when the base (n) is the same in both terms.

Simplify log 2 72 – log 2 9 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 = 8 So log 2 72 – log 2 9 = log 2 8 (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 log 2 8 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…. log 2 8 = x 8 = 2 x And so x = 3 A better ans then is: log 2 72 – log 2 9 = log 2 8 = 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 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! 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 2log log 3 4 Solution First use Law #4 to shift the coefficients (2 & 3) up to the power position 2log log 3 4 = log 3 (5 2 ) + log 3 (4 3 ) which is log 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) = log 3 (25 x 64) Work it out = log 3 (1600) At this stage, check if there is any recognisable power connection between 3 and Maybe check powers of 3 on the calc. There appears to be no connection, so leave this as the ans.

Simplify 2 log log 5 15 Solution Again use Law #4 to shift the coefficients (2 & 2) up to the power position 2log log 5 15 = log 5 (3 2 ) - log 5 (15 2 ) which is log log As this is now of the format log a - log b we can use Law #2 to combine together and get log (a / b) = log 5 (9 / 225) Work it out = log 5 (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 log 5 (1/25) Rewrite as log 5 (5 -2 ) Use Law #4 to drop the power down the front = -2 log 5 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 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. Now use Law #4 to drop powers (2 & 3) to the front Now use Law #5 Ans!

Solution First use Law 3 to change the left side and kill the fraction Now use Law #6 log a 1 = 0 If, find the value of a Now remember 3  3 = 3 x 3 1/2 = 3 3/2 Now Law#4 to bring power down front which is now of form a log 4 3 so a = -3/2

Express in simplest form 4 – 3log 10 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…

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 = and so 4 = log So back to the original question 4 – 3 log 10 x can now be rewritten as log – log 10 x 3 using Law #4 to move the 3 = log 10 (10000 /x 3 ) using Law #2

Simplify 5log Solution Remember to first look for a connection between the 2 and 8 ? As 8 = 2 3 we can write log 2 8 = log 2 (2 3 ) 5log = 5log 2 (2 3 ) + 3 = 5 x 3 log Use Law #4 to move the 3 to front = 5 x 3 (1) + 3 Use Law #5 to simplify log 2 2 =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 log 2 x = 5 so x = 2 5 x = 32 Use LOG LAW #1: Transformation Rule log a n = t n = a t Easy!!

Solve log 3 (1/9) = x Solution log 3 (1/9) = x so 1/9 = 3 x x = -2 Use LOG LAW #1: Transformation Rule log a n = t n = a t i.e = 3 x 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 log 10 of BOTH SIDES then use LAW #4. 3 x = 20 log 10 (3 x ) = log First, take logs 10 of both sides Now use Law#4 on left expression x log 10 3 = log Finally divide both sides by log 10 3 to make x the subject