Exponential functions y=ax

What do they look like ?

y= 2x looks like this

Y=2x

x -1 0 1 2 3

y 0.5 1 2 4 8

Y=10x looks like this

Y=2xY=10x

Y=3x looks like this

Y=2x

Y=10x

Y=3x

y=ex looks like this

y=10x

y=3x

y=2x

y=ex

“e” is a special number in maths,It’s value is

2.718281828.

We will explain the importance of the number e in a later lesson!!

xxx e 32

All these exponential functions have inverses

To find INVERSE We reflect the function in the line y=x

y=10x and y=ex are the most important

y=10xy=ex

The inverse functions are called Logarithms

y=ln(x)y=log(x)

In General for y=ax

FunctionF(x) = 10x

ex

ax

a is any constant

INVERSEF-1(x) = Log10(x) Loge(x) Loga(x)

Remember ff-1(x) = f-1f(x) = x

Log10(x) is written as simply Log(x)Loge(x) is written as Ln(x) Natural or Naperian Log

xFF

x

OF

x

)(

)10(log1

10

So what ?Logarithms allow us to solve equations involving exponentials like :

10X=4 where x is the power

10X=4 eX=4 aX=4

Log(10X)=Log(4) Ln(eX)=Ln(4) Loga(aX)=Loga(4)

X= Log(4) X= Ln(4) X=Loga(4)

Take logs of both sides

Because we are taking ff-1(x)

FUNCTION ax

(EXPONENTIAL)INVERSE FUNCTION (LOG)

So if 10x=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

yxxy

aaa

aaa

yxyx

loglog)(log

.

Indices

Logs

Log Laws – Rule 1

Log Laws – Rule 2

yxy

x

aa

a

aaa

yxx

y

logloglog

Indices

Logs

Log Laws – Rule 3

xkx ak

a loglog

xkx ak

a aa loglog xkk aax log kxk aax log kk xx kk xx

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

1log aa aaa log

1log aa

Why? This equals a1 Because ff-1(x)

Log Laws – Rule 5

01log aAll logs pass through (1,0)

1a

Log laws - Rule 6

xLogx

Log aaa log11

xx

Log aa log1

SO

Using law 2

because Loga1=0

Log laws - Rule 7

a

bbLog

c

ca log

log

The change of base rule Why?

bylet alogby aaa log

ba y ba c

yc loglog Take Logs of both sides

bay cc loglog Using Log Law 3

a

by

c

c

log

log

a

bb

c

ca log

loglog

BUT y=logab

All together

yxxy aaa loglog)(log

yxy

xaaa logloglog

xkx ak

a loglog

1log aa 01log a

xx

Log aa log1

a

bbLog

c

ca log

log

What now1- 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 ax=b4- And now you know why!!

Because they undo the exponential ax ; as they are it’s Inverse :

xaLogSOxLogxfaxf xaa

x )(;)( 1

Next we will use logarithms