# Exponential Growth: Time

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

## Formula

### Formula

Recall that

the exponential change formula is

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

When finding the time t,

switch both sides

and use the 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

### Example

### Solution

The initial value of the population is 10,000.

So A_{0} = 10000.

It says

after how many years will the population

be more than 24000?

So set A = 24000.

The population increases

at a rate of 8% per year.

So r = 0.08/year.

A_{0} = 10000

A = 24000

r = 0.08

Then 10000(1 + 0.08)^{t} = 24000.

The goal is to find the time t.

Divide both sides by 10000.

1 + 0.08 = 1.08

Then 1.08^{t} = 2.4.

log 2.4 and log 1.08 are given.

So common log both sides.

log 1.08^{t} = log 2.4

log 1.08^{t} = t log 1.08

Logarithm of a Power

It says

assume log 2.4 = 0.380, log 1.08 = 0.033.

Then t⋅0.033 = 0.380.

Divide both sides by 0.033.

Move the decimal points

3 digits to the right.

0.380/0.033 = 380/33

Find the value of 380/33

to the ones.

380/33 = 11.xx

t = 11.xx

Round this up to the nearest ones:

11.xx → 12.

The unit of the time is [year].

So write

After 12 years.

t = 11.xx means

after 11.xx years,

the population will be exactly 24,000.

So after 12 years,

the population will be more than 24,000.

So

after 12 years

is the answer.