# Exponential Decay (Part 2)

How to find the amount of time for the exponential decay by using logarithm: formulas, examples, and their solutions.

## Formula

Recall the exponential change formula:*A* = *A*_{0}(1 + *r*)^{t}.

Exponential Decay - Final Value

By using the logarithm,

now you can find the value of *t*.

## Example 1

The initial weight is 100g.

So *A*_{0} = 100.

The final weight is 10g.

So *A* = 10.

The rate is 14% per week,

decreasing [weekly].

So -*r* = -0.14/week.

*A*_{0} = 100*A* = 10

-*r* = -0.14/week

So 100⋅(1 - 0.14)^{t} = 10.

Divide both sides by 100.

And 1 - 0.14 = 0.86.

Then 0.86^{t} = 1/10.

Log both sides.

(base: 0.86)

Then Then *t* = log_{0.86} 1/10.

Logarithmic form

log_{0.86} 1/10 = (log 1/10)/(log 0.86)

Change the base formula

1/10 = 10^{-1}

Negative exponent

log 10^{-1} = -1 log 10

Logarithm of a power

log 10 = 1

Logarithm of the base

log 0.86 = -0.066

Cancel the (-) signs

on both of the numerator and the denominator.

Then (given) = 1.000/0.066.

Move the decimal points of both numbers

3 digits to the right.

(= Multiply 1000

to both of the numerator and the denominator.)

Reduce 1000 to 500

and reduce 66 to 33.

500/33 = 15.xx

So *t* = 15.xx.

*t* = 15.xx weeks.

But *t* should be a natural number.

So round 15.xx up to the nearest ones:

15.xx → 16.

So *t* = 16 weeks.

So the answer is [after 16 weeks].

## Formula: Continuous Decay

Recall the continuous exponential change formula:*A* = *A*_{0}*e*^{rt}.

Continuous Exponential Decay - Final Value

By using the logarithm,

you can also find *t* from this formula.

## Example 2

The half-life is the amount of time

that takes to change

from *A*_{0} to (1/2)*A*_{0}.

So set *A*_{0} = *A*_{0}

and *A* = *A*_{0}/2.

The rate is 5% per minute,

decreasing [minutely].

So -*r* = -0.05/minute.

*A*_{0} = *A*_{0}*A* = *A*_{0}/2

-*r* = -0.05/minute

Then *A*_{0}⋅*e*^{-0.05⋅t} = *A*_{0}/2.

Divide both sides by *A*_{0}.

Then *e*^{-0.05t} = 1/2.

Natural log both sides.

Then -0.05*t* = ln (1/2).

Natural logarithm

1/2 = 2^{-1}

ln 2^{-1} = -1 ln 2

Logarithm of a power

ln 2 = 0.69

-1⋅0.69 = -0.69

Cancel the (-) signs in both sides.

Move the decimal points of both sides

2 digits to the right.

(= Multiply 100 to both sides.)

Divide both sides by 5.

Then *t* = 69/5.

Then the half-life of the substance is

69/5 minutes.

(= 13.8 minutes)

When finding the half-life,

the answer don't have to be a natural number.

So you don't have to

round the answer up to the nearest ones.