The function ex and its inverse, lnxThe functions like y = 2x, y = 3x are called exponential functions because the variable x is the power (exponent or index) of a base number.

The graph of y = 2x, y = 3x and y = ex

0

2

4

6

8

10

-4 -3 -2 -1 0 1 2 3 4

y=2x

y=ex

y=3x

x

It is clear from the graph that the number e is somewhere between 2 and 3 but closer to 3 than 2.This is a special number and its value correct to 8 decimal places is e =2.18281828.

Graph of ex and lnx

y = ex

y = lnx

y = x

x

y

1

1

y = ex 0Domain x and Range y

y = lnx 0Domain x and Range x

Evaluating function ex and lnxEvaluate

(i) e2 (ii) e-3 (iii) ln0.5 (iv) ½ ln10

(i) 7.39 (ii) 0.0498 (iii) 7.39 (iv) 1.15

(i) ln ex = 3 (ii) elnx = 5 (iii) e2lnx = 16 (iv) e-lnx = ½

(i) x = 3 (ii) x = 5 (iii) x = 4 (iv) x = 2

Find the value of x

Solving equations involving ex and lnxSolve for x

(i) 3e2x – 1 = 54

3e2x = 55

e2x = 18.333..

2x= ln18.333..

2x= 2.9087..

x= 1.45

(ii) 3e2x –5ex = 2

Let y = ex

3y2 – 5y = 2

(3y )(y )

3y2 – 5y – 2 = 0

(3y + 1)(y - 2)

y = - 1/3 or y = 2

ex = 2 x = 0.693

Solving equations involving ex and lnx

Solve for x

(i) ln(3x – 5) = 3.4

3x – 5 = e3.4

3x – 5 = 29.964…

3x = 34.964…

x= 11.7.

(ii) ln(3x + 1) – ln3 = 1

ln((3x + 1)/3) = 1

((3x + 1)/3) = 2.718…

((3x + 1)/3) = e1

3x + 1 = 8.1548..

3x = 7.1548…

x = 2.38

Exponential Decay/Growth

-0.2tN=50e

A quantity N is decreasing such that at time t

(a) Find the value of N when t = 5

(b) Find the value of t when t = 2

-0.2 5N=50e (a)

-0.22=50e t(b)

= 18.4

0 2125

- . te 125ln =-0.2t

t = 16.1