Logarithms 28 August, 2015F L1 MH Objectives : To know what log means To learn the laws of logs To simplify logarithmic expressions To solve equations of the type a x =b

Reasons for Studying this 28 August, 2015F L1 MH We WILL meet the graph of y=a x 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 = a x

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

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

Logarithm 28 August, 2015F L1 MH

logarithms log 2 64=6because 2 6 = 64 log 2 (1/2)=-1because 2 -1 = ½ log 2 1=0because 2 0 = 1 log 2 √2=1/2because 2 1/2 = √2 28 August, 2015F L1 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: ChemistrypH value of a liquid Physicspower 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 = 1 Scientists only work with 2 specific bases (log, ln) We do not write the 10 as Log means to base 10 Log10=1Log 0.1 = -1 Log1=0 Log100=2Log 0.01=-2 Log1000=3 Log0.001=-3 We can not take logs of negative numbers 28 August, 2015F L1 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 e x, (we will see this later) e is a very important irrational number in maths and science, it has some very special properties!! 28 August, 2015F L1 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” 28 August, 2015F L1 MH log index Log 4 = (3 sig fig) – on calculator

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 28 August, 2015F L1 MH

Some Important Rules 28 August, 2015F L1 MH These are the Laws of Logs a 0 =1 a 1 =a

Using these Rules- Simplify 28 August, 2015F L1 MH ~ Log a 6

Simplify 28 August, 2015F L1 MH Express in terms of log a x, log a y, log a z Log a = log a x 3 – log a y 2 z Log a = log a x 3 – (log a y 2 + log a z) Log a = 3log a x – 2log a y - log a z)

28 August, 2015F L1 MH Do Ex 11A Page 325 q 1 - 5

We don’t actually take the logs anywhere: we put them in, but the process is always called taking logs! Solving e.g.1 SolveSolution: We “take” logs ( Notice that 2 < x < 3 since ) 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.

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

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

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

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. ) Exercises

Do Ex 11A page 326 no 6 ff 28 August, 2015F L1 MH