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

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

Change of Base WE only have log 10 and ln (loge)on our calculators BUT We can calculate the log to any base logx by rewriting the base This is called changing the base * F L 1 MH

Change of base rule If y = logab Then ay = b Taking logs of both sides gives logc ay = logcb (c can be any base number) So ylogc a = logcb ( laws of logs ) So y = logcb/ logca (divide by logca) Therefore * F L 1 MH

Example Calculate log 47 to 3 sig fig Log 47 = log 107 / log 104 (Change of base) = Can someone work this out on their Please ! * F L 1 MH

A very IMPORTANT result From the change of base rule we can say And of course Logyy=1 SO * F L 1 MH

What does f(x)=logx look like red is to base e, green is to base 10, purple is to base 1. ALL Pass through (1, 0) * F L 1 MH

Exponential Functions The inverse to f(x)=logax 10(Log 10 x) is the same as x And generally a(Logax) is the same as x So f(x)= log 10(x) and f(x)= 10 x are inverse functions. One undoes the other * F L 1 MH

More formally find f-1(x) of f(x)=logx Step 1: Step 2: Let y=logax Rearrange in terms of x (To do this raise both sides to the power of a ) ay = alogax -> ay = x Step 3 : Swap x and y -> y = ax * F L 1 MH If f(x)=logax then f-1(x) = ax

Exercise - Task 1. Neatly draw the graph of f(x)=ax 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 ex 6. Try and guess (by considering some points the gradient of ex (at SAY x=-1, 0, 1 or x= 0, 1, 2) * F L 1 MH

f(x)=ax * F L 1 MH

f(x)=lnx; f(x)=ex * F L 1 MH