Exponential Decay: Time

How to find the time of exponential decay: formula, 1 example, and its solution.

Formula

Formula

Just like finding the time of exponential growth,
to find the time of exponential decay,
use the exponential change formula.

A0(1 + r)t = A

A0: Initial value
r: Rate of change (per time period)
t: Number of time period
A: Final value

Example

Example

Solution

The initial value of the weight is 100g.

So A0 = 100g.

It says
after how many weeks will the weight
be less than 10g?

So set A = 10g.

The weight decreases
at a rate of 14% per week.

So r = -0.14/week.

The minus sign means decreasing.

A0 = 100
A = 10
r = -0.14

Then 100⋅(1 - 0.14)t = 10.

The goal is to find the time t.

Divide both sides by 100.
1 - 0.14 = 0.86

Then 0.86t = 0.1.

log 0.86 is given.

So common log both sides.

log 0.86t = log 0.1

0.1 = 1/10 = 10-1

Negative Exponent

log 0.86t = t log 0.86

log 10-1 = -1 log 10

Logarithm of a Power

It says
assume log 0.86 = -0.066.

log 10 = 1

Logarithm of Itself

So t⋅(-0.066) = -1⋅1.

Multiply -1 to both sides.

Then 0.066t = 1.

Divide both sides by 0.066.

Move the decimal points
3 digits to the right.

1/0.066 = 1000/66

Find the value of 1000/66
to the ones.

1000/66 = 15.xx

t = 15.xx
Round this up to the nearest ones:
15.xx → 16.

The unit of the time is [week].

So write
After 16 weeks.

t = 15.xx means
after 15.xx weeks,
the weight will be exactly 10g.

So after 16 weeks,
the weight will be less than 10g.

So
after 16 weeks
is the answer.