Compound Interest: Time

A compounded interest means
add the principle and the interest,
calculate the next interest,
and repeating this process.

The amount of money shows exponential growth.

To find the time
of a compound interest investment,
use the exponential growth formula:
A₀(1 + r)^t = A.

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

Compound Interest: Final Value

The initial value of the investment is $1,000.

So A₀ = $1000.

It says
after how many years will the investment
worth more than $1,800?

So set A = $1800.

The investment is at a rate of 6% per year.

So r = 0.06/year.

A₀ = 1000
A = 1800
r = 0.06/year

The investment is compounded yearly.

Then 1000(1 + 0.06)^t = 1800.

The goal is to find the time t.

Divide both sides by 1000.
1 + 0.06 = 1.06

Then 1.06^t = 1.8.

log 1.8 and log 1.06 are given.

So common log both sides.

log 1.06^t = log 1.8

log 1.06^t = t log 1.06

Logarithm of a Power

It says
assume log 1.8 = 0.255, log 1.06 = 0.025.

Then t⋅0.025 = 0.255.

Divide both sides by 0.025.

Move the decimal points
3 digits to the right.

0.255/0.025 = 255/25

Find the value of 255/25
to the ones.

255/25 = 10.xx

t = 10.xx
Round this up to the nearest ones:
10.xx → 11.

The unit of the time is [year].

So write
After 11 years.

t = 10.xx means
after 10.xx years,
the investment will worth exactly $1,800.

So after 11 years,
the investment will worth more than $1,800.

So
after 11 years
is the answer.