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# Exponential

See how to solve an exponential equation/inequality/function/change.
13 examples and their solutions.

23x - 1 = 4
Solution

3x - 5 = 94
Solution

74x + 8 = 1
Solution

125⋅5x = (125)x
Solution

4x = 3⋅2x + 4
Solution

25x - 9 > 4x
Solution

116(18)x (14)x
Solution

## Exponential Function: Graph

### Graph: y = ax(a > 1)

1. The graph passes (0, 1).
(0, a0) = (0, 1)
2. The asymptote of the graph is the x-axis.
(= The graph follows the x-axis.)

### Graph: y = ax(0 < a < 1)

1. The graph passes (0, 1).
(0, a0) = (0, 1)
2. The asymptote of the graph is the x-axis.
(= The graph follows the x-axis.)

Graph y = 2x.
Solution

Graph y = (13)x.
Solution

## Exponential Growth/Decay: Final Value

### Formula

A = A0(1 + r)t
A: Final value
A0: 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 1.0712 = 2.252.)
Solution

### Example

A radioactive substance weighs 80g. If it decreases at a rate of 5% per minute, find the expected weight 1 hour later.
(Assume 0.9560 = 0.046.)
Solution

## Continuous Exponential Growth/Decay: Final Value

### Formula

A = A0ert
A: Final value
A0: 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 e1.8 = 6.05.)
Solution

### 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-3 = 0.050.)
Solution