1. Logarithms 25 January, 2016F 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

2. 1. Simplify the following: (a) (c) (b) (d) (a) 1 Ans: (b) 0 (c) 19 (d) b Exercises

3. Change of Base WE only have log 10 and ln (log e )on our calculators BUT We can calculate the log to any base log x by rewriting the base This is called changing the base 25 January, 2016F L1 MH

4. Change of base rule If y = log a b Then a y = b Taking logs of both sides gives log c a y = log c b (c can be any base number) So ylog c a = log c b ( laws of logs ) So y = log c b/ log c a (divide by log c a) Therefore 25 January, 2016F L1 MH

5. Example Calculate log 4 7 to 3 sig fig Log 4 7 = log 10 7 / log 10 4 (Change of base) = Can someone work this out on their Please ! 25 January, 2016F L1 MH

6. A very IMPORTANT result From the change of base rule we can say And of course Log y y=1 SO 25 January, 2016F L1 MH

7. What does f(x)=logx look like 25 January, 2016F L1 MH red is to base e, green is to base 10, purple is to base 1. ALL Pass through (1,0)

8. Exponential Functions 25 January, 2016F L1 MH The inverse to f(x)=log a x 10 ( Log 10 x ) is the same as x And generally a ( Log a x ) is the same as x So f(x)= log 10 (x) and f(x)= 10 x are inverse functions. One undoes the other

9. More formally find f -1 (x) of f(x)=logx 25 January, 2016F L1 MH Step 1: Let y=log a x Step 2: Rearrange in terms of x (To do this raise both sides to the power of a ) a y = a log a x -> a y = x Step 3 : Swap x and y -> y = a x If f(x)=log a x then f -1 (x) = a x

10. Exercise - Task 1. Neatly draw the graph of f(x)=a x for these values of a ; 1,2,3. (On graph paper neatly use calculator) 2. Choose your domain to be -4 ≤x ≤3 3. Measure the gradient at Pt(0,1) carefully 4. Guess which value of a gives a gradient of 1 at (0,1) 5. Draw on graph paper f(x)=lnx and e x 6. Try and guess (by considering some points the gradient of e x (at SAY x=-1, 0,1 or x= 0,1,2) 25 January, 2016F L1 MH

11. f(x)=a x 25 January, 2016F L1 MH

12. f(x)=lnx; f(x)=e x 25 January, 2016F L1 MH