Step by step
The question
Question: How long until $3000 invested at 5% compounded monthly grows to $4500?
1 Write the equation
The compound interest rule, with r/n = 0.05 divided by 12: 0.004167.
We know the start, $3000, and the end, $4500.
t is the unknown, up in the power.
2 Get the power alone
To remove times 3000, we divide by 3000.
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.
3000 divided by 3000 is 1.
So just the power is left.
4500 divided by 3000 is 1.5.
Now we have the bracket to the power 12t = 1.5.
Careful!
When the power comes down, all of it comes down: 12t, not just t.
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, 12t, comes down in front.
5 Get t alone
Divide both sides by 12 times log 1.004167.
So t is alone.
6 Use a calculator
With a calculator: log 1.5 is about 0.176091.
And log 1.004167 is about 0.0018058.
So t is about 8.13 years, to two decimal places.
7 Check
Let's check: put t = 8.13 back in.
With a calculator.
We get about $4500.86.
That is 4500, 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.