# Exponential Decay: Time

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

## Formula

Just like finding the time of exponential growth,

to find the time of exponential decay,

use the exponential change formula.

A_{0}(1 + r)^{t} = A

A_{0}: Initial value

r: Rate of change (per time period)

t: Number of time period

A: Final value

## Example

The initial value of the weight is 100g.

So A_{0} = 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.

A_{0} = 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 0.86^{t} = 0.1.

log 0.86 is given.

So common log both sides.

log 0.86^{t} = log 0.1

0.1 = 1/10 = 10^{-1}

Negative Exponent

log 0.86^{t} = 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.