Ximpledu

Exponential

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

Exponential Equation

Example

23x - 1 = 4
Solution

Example

3x - 5 = 94
Solution

Example

74x + 8 = 1
Solution

Example

125⋅5x = (125)x
Solution

Example

4x = 3⋅2x + 4
Solution

Exponential Inequality

Example

25x - 9 > 4x
Solution

Example

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.)

Example

Graph y = 2x.
Solution

Example

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

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