Logarithms Objectives To know what log means To

Logarithms Objectives : • To know what log means • To learn the laws of logs • To simplify logarithmic expressions • To solve equations of the type * F L 1 MH ax=b

Reasons for Studying this 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 * F L 1 MH

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

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 This index is called a logarithm ( or log ) and 10 is between 10 and 100 ( or ), x the base. lies between 1 and 2. Our calculators give us the value of the logarithm of 75 with a base of 10. The button is marked Log 75 The value is so, ( 3 d. p. )

Logarithm * F L 1 MH

logarithms * log 264=6 because 26 = 64 log 2(1/2)=-1 because 2 -1 = ½ log 21=0 because 20= 1 log 2√ 2=1/2 because 21/2 = √ 2 F L 1 MH

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 Physics p. H value of a liquid 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 Log 10 10 = 1 Scientists only work with 2 specific bases (log, ln) We do not write the 10 as Log means to base 10 Log 10=1 Log 0. 1 = -1 Log 1=0 Log 100=2 Log 0. 01=-2 Log 1000=3 Log 0. 001=-3 We can not take logs of negative numbers * F L 1 MH

Log is an Inverse Log is the inverse to 10 x (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!! * F L 1 MH

So A logarithm is just an index Solve the equation 10 x = 4 giving the answer correct to 3 significant figures. “x is the logarithm of 4 with a base of 10” index log Log 4 = 0. 602 (3 sig fig) – on calculator * F L 1 MH

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 * F L 1 MH

Some Important Rules a 0=1 a 1=a These are the Laws of Logs * F L 1 MH

Using these Rules- Simplify ~ Loga 6 * F L 1 MH

Simplify Express Loga * in terms of logax, logay, logaz = loga x 3 – logay 2 z = loga x 3 – (logay 2 + logaz) = 3 loga x – 2 logay - logaz) F L 1 MH

Do Ex 11 A Page 325 q 1 - 5 * F L 1 MH

Solving e. g. 1 Solve ( Notice that 2 < x < 3 since ) We “take” logs Solution: 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 A 2 ).

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 Ø The Definition of a Logarithm Ø The “Power to the Front” law of logs: Ø 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 correct to 2 d. p. giving the answer Solution: Divide by 200: Take logs: Power to the front: Rearrange: ( 2 d. p. )

Do Ex 11 A page 326 no 6 ff * F L 1 MH