# Exponential

See how to solve an exponential equation/inequality/function/change.

13 examples and their solutions.

## Exponential Equation

### Example

2

Solution ^{3x - 1}= 4 2

2

3x - 1 = 2 - [1]

3x = 3

x = 1

^{3x - 1}= 42

^{3x - 1}= 2^{2}3x - 1 = 2 - [1]

3x = 3

x = 1

[1]

2

The bases are the same: 2.

Then the exponents are the same.

^{3x - 1}= 2^{2}The bases are the same: 2.

Then the exponents are the same.

Close

### Example

3

Solution ^{x - 5}= 9^{4} 3

3

= 3

x - 5 = 8

x = 13

^{x - 5}= 9^{4}3

^{x - 5}= (3^{2})^{4}= 3

^{8}- [1]x - 5 = 8

x = 13

[1]

Close

### Example

7

Solution ^{4x + 8}= 1 7

= 7

4x + 8 = 0

4x = -8

x = -2

^{4x + 8}= 1= 7

^{0}4x + 8 = 0

4x = -8

x = -2

Close

### Example

125⋅5

Solution ^{x}= (125)^{x} 125⋅5

5

5

5

x + 3 = -2x

3x = -3

x = -1

^{x}= (125)^{x}5

^{3}⋅5^{x}= 25^{-x}5

^{3 + x}= (5^{2})^{-x}5

^{x + 3}= 5^{-2x}x + 3 = -2x

3x = -3

x = -1

Close

### Example

4

Solution ^{x}= 3⋅2^{x}+ 4 4

4

(2

(2

1) 2

2

= 2

x = 2

2) 2

2

x = 2

^{x}= 3⋅2^{x}+ 44

^{x}- 3⋅2^{x}- 4 = 0(2

^{x})^{2}- 3⋅2^{x}- 4 = 0(2

^{x}- 4)(2^{x}+ 1) = 0 - [1]1) 2

^{x}- 4 = 02

^{x}= 4= 2

^{2}x = 2

2) 2

^{x}+ 1 = 02

^{x}= -1( x ) - [2]x = 2

[2]

2

The left side, 2

The right side, -1, is (-).

So there's no solution for case 2.

^{x}= -1The left side, 2

^{x}, is (+).The right side, -1, is (-).

So there's no solution for case 2.

Close

## Exponential Inequality

### Example

2

Solution ^{5x - 9}> 4^{x} 2

2

5x - 9 > 2x

3x > 9

x > 3

^{5x - 9}> 4^{x}2

^{5x - 9}> 2^{2x}- [1]5x - 9 > 2x

3x > 9

x > 3

[1]

The base 2 is in

2 > 1.

So the order of the inequality sign

doesn't change.

> → >

2 > 1.

So the order of the inequality sign

doesn't change.

> → >

Close

### Example

116⋅(18)

Solution ^{x}≤ (14)^{x} 116⋅(18)

(12)

(12)

4 + 3x ≥ 2x - [2]

x ≥ -4

^{x}≤ (14)^{x}(12)

^{4}⋅(12)^{3x}≤ (12)^{2x}(12)

^{4 + 3x}≤ (12)^{2x}- [1]4 + 3x ≥ 2x - [2]

x ≥ -4

[2]

The base 1/2 is in

0 < 1/2 < 1.

So the order of the inequality sign

does change.

≤ → ≥

0 < 1/2 < 1.

So the order of the inequality sign

does change.

≤ → ≥

Close

## Exponential Function: Graph

### Graph: y = a^{x} (a > 1)

(0, a

^{0}) = (0, 1)

2. The asymptote of the graph is the x-axis.

(= The graph follows the x-axis.)

### Graph: y = a^{x} (0 < a < 1)

(0, a

^{0}) = (0, 1)

2. The asymptote of the graph is the x-axis.

(= The graph follows the x-axis.)

### Example

Graph y = 2

Solution ^{x}.Draw (0, 1).

↓

Draw the points

when x = 1, 2, 3 and -1, -2, -3.

(1, 2

(2, 2

(3, 2

(-1, 2

(-2, 2

(-3, 2

when x = 1, 2, 3 and -1, -2, -3.

(1, 2

^{1}) = (1, 2)(2, 2

^{2}) = (2, 4)(3, 2

^{3}) = (3, 8)(-1, 2

^{-1}) = (-1, 1/2)(-2, 2

^{-2}) = (-2, 1/4)(-3, 2

^{-3}) = (-3, 1/8)↓

Close

### Example

Graph y = (13)

Solution ^{x}.↓

Draw the points

when x = 1, 2 and -1, -2.

(1, (1/3)

(2, (1/3)

(-1, (1/3)

(-2, (1/3)

when x = 1, 2 and -1, -2.

(1, (1/3)

^{1}) = (1, 1/3)(2, (1/3)

^{2}) = (2, 1/9)(-1, (1/3)

^{-1}) = (-1, 3)(-2, (1/3)

^{-2}) = (-2, 3^{2}) = (-2, 9)↓

Close

## Exponential Growth/Decay: Final Value

### Formula

A = A

A: Final value_{0}(1 + r)^{t}A

_{0}: Initial value

r: Rate of change

t: Time

Compound Interest

Exponential Growth/Decay: Time

### Example

The population of a town is 10,000. If it increases at a rate of 7% per year, find the expected population 12 years later.

(Assume 1.07

Solution (Assume 1.07

^{12}= 2.252.) A

r = 0.07 /year

t = 12 years

A = 10000⋅(1 + 0.07)

= 10000⋅1.07

= 10000⋅2.252

= 22520

_{0}= 10000r = 0.07 /year

t = 12 years

A = 10000⋅(1 + 0.07)

^{12}= 10000⋅1.07

^{12}= 10000⋅2.252

= 22520

Close

### Example

A radioactive substance weighs 80g. If it decreases at a rate of 5% per minute, find the expected weight 1 hour later.

(Assume 0.95

Solution (Assume 0.95

^{60}= 0.046.) A

r = -0.05 /minute

t = 60 minutes

A = 80⋅(1 - 0.05)

= 80⋅0.95

= 80⋅0.046

= 3.68 g

_{0}= 80 gr = -0.05 /minute

t = 60 minutes

A = 80⋅(1 - 0.05)

^{60}= 80⋅0.95

^{60}= 80⋅0.046

= 3.68 g

Close

## Continuous Exponential Growth/Decay: Final Value

### Formula

A = A

A: Final value_{0}e^{rt}A

_{0}: Initial value

e: Constant number (= 2.71828...)

r: Rate of change

t: Time

Compound Interest

Continuous Exponential Growth/Decay: Time

Constant e

### Example

A substance weighs 10g. If it continuously increases at a rate of 3% per second, find the expected weight 1 minute later.

(Assume e

Solution (Assume e

^{1.8}= 6.05.) A

r = 0.03 /second

t = 60 seconds

A = 10⋅e

= 10⋅e

= 10⋅6.05

= 60.5 g

_{0}= 10r = 0.03 /second

t = 60 seconds

A = 10⋅e

^{0.03⋅60}= 10⋅e

^{1.8}= 10⋅6.05

= 60.5 g

Close

### Example

A radioactive substance weighs 80g. If it continuously decreases at a rate of 5% per minute, find the expected weight 1 hour later.

(Assume e

Solution (Assume e

^{-3}= 0.050.) A

r = -0.05 /minute

t = 60 minutes

A = 80⋅e

= 80⋅e

= 80⋅0.05

= 4.0 g

_{0}= 80r = -0.05 /minute

t = 60 minutes

A = 80⋅e

^{-0.05⋅60}= 80⋅e

^{-3}= 80⋅0.05

= 4.0 g

Close