Exponential Equations

Do you remember how to solve 2x = 16?

2x = 16

An equation with unknown indices is called an exponential equation.

2x = 24

x = 4

Example of an Exponential equation

Express both sides as

powers of 2

How to solve 22x + 2x 6 = 0?

Since 22x = (2x)2, we can transform the original

equation into a quadratic equation in 2x.

Let me try to solve the equation.

22x + 2x 6 = 0

(2x)2 + 2x 6 = 0

Let y = 2x, the equation becomes

y2 + y 6 = 0(y 2)(y + 3) = 0

y = 2 or y = 3

2x = 2 or 2x = 3 (rejected)

2x = 21

x = 12x is positive.

Follow-up question

22x + 4(2x) 32 = 0

Let y = 2x, the equation becomesy2 + 4y 32 = 0

(y 4)(y + 8) = 0y = 4 or y = 8

2x = 22

x = 2

2x = 4 or 2x = 8 (rejected)

Solve 22x + 4(2x) 32 = 0.

(2x)2 + 4(2x) 32 = 0

How to solve the following simultaneous

equations?

By writing 81 = 34 and 4 = 22, the simultaneous exponential equations

can be simplified.

3x + y = 81 ……(1)2x + 5y = 4 ……(2)

Let me try to solve the equations. From (1), 3x + y = 34

From (2), 2x + 5y = 22

3x + y = 81 ……(1)2x + 5y = 4 ……(2)

x + y = 4

x + 5y = 2

x + y = 4 ……(3)x + 5y = 2 ……(4)

The simultaneous equations become:

Solving (3) and (4), we have

1and 3 yx

Follow-up question

From (1), 2x + 2y = 20

2x + 2y = 1 ……(1)

5x + 3y = ……(2)

x + 2y = 0 ……(3)x + 3y = 1 ……(4)

The simultaneous equations become:

Solving (3) and (4), we have1and 2 yx

Solve the following simultaneous equations:

5

1

x + 2y = 0

From (2), 5x + 3y = 51 x + 3y = 1