Exponential functions y=a x What do they look like ? y= 2 x looks like this
Y=2 x x 0123 y
Y=10 x looks like this Y=2 x Y=10 x
Y=3 x looks like this Y=2 x Y=10 x Y=3 x
y=e x looks like this y=10 x y=3 x y=2 x y=e x “e” is a special number in maths, It’s value is We will explain the importance of the number e in a later lesson!!
All these exponential functions have inverses To find INVERSE We reflect the function in the line y=x
y=10 x and y=e x are the most important y=10 x y=e x The inverse functions are called Logarithms y=ln(x) y=log(x)
In General for y=a x Function F(x) =10 x e x a x a is any constant INVERSE F -1 (x) =Log 10 (x)Log e (x)Log a (x) Remember ff -1 (x) = f -1 f(x) = x Log 10 (x) is written as simply Log(x) Log e (x) is written as Ln(x) Natural or Naperian Log
So what ? Logarithms allow us to solve equations involving exponentials like : 10 X =4 where x is the power 10 X =4e X =4a X =4 Log(10 X )=Log(4)Ln(e X )=Ln(4)Log a (a X )=Log a (4) X= Log(4)X= Ln(4)X=Log a (4) Take logs of both sides Because we are taking ff -1 (x) FUNCTION a x (EXPONENTIAL) INVERSE FUNCTION (LOG)
So if 10 x =4 then x=Log(4) The power “x” is therefore a logarithm !! Logarithms are powers in disguise !! And so the laws of logs are a little like the laws of indices
Indices Logs Log Laws – Rule 1 Log Laws – Rule 2 Indices Logs
Log Laws – Rule 3 Why? Rise both sides to power a Use the laws of indices on RHS RHS ff -1 (x)=x This is perhaps the most useful Rule LHS ff -1 (x)=x
Log Laws – Rule 4 Why? Rise both sides to power a Log Laws – Rule 5 All logs pass through (1,0)
Log laws - Rule 6 SO Using law 2 because Log a 1=0
Log laws - Rule 7 The change of base rule Why? Take Logs of both sides Using Log Law 3 BUT y=log a b
All together
What now 1- The laws of logarithms are given to you in an exam, you don’t have to remember them 2- But you do have to use them 3- We use logarithms to solve things like a x =b 4- And now you know why!! Because they undo the exponential a x ; as they are it’s Inverse : Next we will use logarithms