Step by step
The question
Question: $2000 grows by 3% a year.
How many years until it is $2600?
1 Write the equation
The compound interest rule, with n = 1: once a year.
We know the start, $2000, and the end, $2600.
t is the unknown, up in the power.
2 Get the power alone
To remove times 2000, we divide by 2000.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
2000 divided by 2000 is 1.
So just the power is left.
2600 divided by 2000 is 1.3.
Now we have 1.03 to the power t = 1.3.
Careful!
Each year it keeps all of it and gets 3% more.
So we multiply by 1.03, not by 0.03.
3 Take the log of both sides
Logs bring a power down.
So we take the log of both sides.
We do it on both sides, so they stay equal.
4 Bring the power down
The power rule for logs: the power comes down in front, as times.
The whole power, t, comes down in front.
5 Get t alone
Divide both sides by log 1.03.
So t is alone.
6 Use a calculator
With a calculator: log 1.3 is about 0.113943.
And log 1.03 is about 0.0128372.
So t is about 8.88 years, to two decimal places.
7 Check
Let's check: put t = 8.88 back in.
With a calculator.
We get about $2600.31.
That is 2600, up to the rounding.
It works!
Let's look back.
First: write the equation.
Then: get the power alone.
Then: take the log of both sides.
Then: bring the power down.
Then: get t alone.
Then: use a calculator.
Last: check.