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 = A0(1 + r)t.

When finding the time t,
switch both sides
and use the 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 population is 10,000.

So A0 = 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.

A0 = 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 1.08t = 2.4.

log 2.4 and log 1.08 are given.

So common log both sides.

log 1.08t = log 2.4

log 1.08t = 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.