46: Indices and Laws of Logarithms “Teach A Level Maths” Vol. 1: AS Core Modules 46: Indices and Laws of Logarithms © Christine Crisp

Unknown Indices We have met the graph of and seen that it represents growth or decay. Because of important practical applications of growth and decay, we need to be able to solve equations of the type where a and b are constants. Equations with unknown indices are solved using logarithms. We will see what a logarithm is and develop some rules that help us to solve equations.

e.g. How would you solve ? Ans: If we notice that

e.g. How would you solve then, (1) becomes - - - - (1) Ans: If we notice that We can use the same method to solve or

log Suppose we want to solve We need to write 75 as a power ( or index ) of 10. Tip: It’s useful to notice that, since 75 lies between 10 and 100 ( or ), x lies between 1 and 2. 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. The button is marked log The value is ( 3 d.p. ) so,

x is the logarithm of 4 with a base of 10 A logarithm is just an index. To solve an equation where the index is unknown, we can use logarithms. e.g. Solve the equation giving the answer correct to 3 significant figures. x is the logarithm of 4 with a base of 10 We write ( 3 s.f. ) In general if then index = log

Memory aid a to the power of x = b In general if then In general if

Exercise Solve the following equations giving the answers correct to 2 d.p. (a) (b) Solution: (a) ( 2 d.p. ) (b) ( 2 d.p. )

In the exercise, we saw that Generalizing this, In general if then This relationship is also true changing from the log form to the index form,

The equation When the base, a, is 10, we found the equation is easy to solve. e.g. Solve the equation Solution: e.g. To solve we could write BUT there are no values for logs with base 2 on our calculators so we can’t find this as a simple number. We need to develop some laws of logs to enable us to solve a variety of equations with unknown indices or logs

A law of logs for ( from the calculator ) e.g. Also, ( from calculator ) And, ( from calculator )

A law of logs for ( from the calculator ) e.g. Also, ( from calculator ) And, ( from calculator )

A law of logs for ( from the calculator ) e.g. Also, ( from calculator ) And, ( from calculator ) We get

A law of logs for The same reasoning holds for any base, a, so ( the “power to the front ” law of logs )

Solving e.g.1 Solve ( Notice that 2 < x < 3 since ) Solution: We “take” logs We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs! Using the “power to the front” law, we can simplify the l.h.s. 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 ).

Solving e.g.2 Solve the equation Solution: We must change the equation into the form before we take logs. Divide by 100: Take logs: Using the “power to the front” law:

SUMMARY The Definition of a Logarithm The “Power to the Front” law of logs: Solving the equation Divide by n “Take” logs Use the power to the front law Rearrange to find x.

Exercises 1. Solve the following equations giving the answers correct to 2 d.p. (a) (b) (a) “Take” logs: ( 2 d.p. ) (b) “Take” logs: ( 2 d.p. )

Exercises 2. Solve the equation giving the answer correct to 2 d.p. Solution: Divide by 200: Take logs: Power to the front: Rearrange: ( 2 d.p. )

Log laws for Multiplying and Dividing We’ll develop the laws by writing an example with the numbers in index form.

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So, In general,

Any positive integer could be used as a base instead of 10, so we get: A similar rule holds for dividing. If the base is missed out, you should assume it could be any base e.g. might be base 10 or any other number.

SUMMARY The Laws of Logarithms are: 1. Multiplication law 2. Division law 3. Power law The definition of a logarithm: leads to 4. 5. 6.

e.g. 1 Express the following in terms of (a) (b) (c) Solution: (a) ( Law 1 ) (b) ( Law 3 ) (c) Either ( Law 2 ) ( Law 4 ) Or ( Law 3 )

e.g. 2 Express in terms of and Solution: We can’t use the power to the front law directly! ( Why not? ) There is no bracket round the ab, so the square ONLY refers to the b. So, ( Law 1 ) ( Law 3 )

e.g. 3 Express each of the following as a single logarithm in its simplest form: (b) Solution: (a) (b) This could be simplified to

Exercise 1. Express the following in terms of (a) (b) (c) Ans: (a) (b) (c) 2. Express in terms of and Ans: 3. Express the following as a single logarithm in its simplest form: (a) (b) (a) (b) Ans:

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

x is the logarithm of 4 with a base of 10 A logarithm is just an index. To solve an equation where the index is unknown, we can use logarithms. e.g. Solve the equation giving the answer correct to 3 significant figures. x is the logarithm of 4 with a base of 10 We write In general if then = log index ( 3 s.f. ) ( from the calculator )

Generalizing this, This relationship is also true changing from the log form to the index form, In the exercise we used logs with a base of 10 but the definition holds for any base, so so, Base

SUMMARY The Definition of a Logarithm Solving the equation “Take” logs The “Power to the Front” law of logs: Use the power to the front law Rearrange to find x. Divide by n

Solving e.g.1 Solve Solution: We “take” logs We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs! 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.

Solution: We must change the equation into the form before we take logs. e.g.2 Solve the equation Using the “power to the front” law: Divide by 100: Take logs: