Logarithms Objectives : To know what log means To learn the laws of logs To simplify logarithmic expressions To solve equations of the type ax=b 8 September, 2018 F L1 MH

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 8 September, 2018 F L1 MH

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 75 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 75 The value is ( 3 d.p.) so,

Logarithm   8 September, 2018 F L1 MH

logarithms log264=6 because 26 = 64 log2(1/2)=-1 because 2-1 = ½ log21=0 because 20= 1 log2√2=1/2 because 21/2 = √2 8 September, 2018 F 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: Chemistry pH value of a liquid Physics 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 Log10 10 = 1 Scientists only work with 2 specific bases (log, ln) We do not write the 10 as Log means to base 10 Log10=1 Log 0.1 = -1 Log1=0 Log100=2 Log 0.01=-2 Log1000=3 Log0.001=-3 We can not take logs of negative numbers 8 September, 2018 F L1 MH

Log is an Inverse Log is the inverse to 10x (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!! 8 September, 2018 F L1 MH

So A logarithm is just an index Solve the equation 10x = 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 8 September, 2018 F L1 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   8 September, 2018 F L1 MH

Some Important Rules These are the Laws of Logs a0=1 a1=a 8 September, 2018 F L1 MH

Using these Rules- Simplify ~ Loga6 8 September, 2018 F L1 MH

Loga = 3loga x – 2logay - logaz) Simplify Express in terms of logax, logay, logaz Loga = loga x3 – logay2z Loga = loga x3 – (logay2 + logaz) Loga = 3loga x – 2logay - logaz) 8 September, 2018 F L1 MH

Do Ex 11A Page 325 q 1 - 5 8 September, 2018 F L1 MH

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. )

Do Ex 11A page 326 no 6 ff 8 September, 2018 F L1 MH