Step by step
The question
Question: A substance has a half-life of 5 years.
How long until 80 g decays to 30 g?
1 Write the equation
The amount left is the start, times (1/2) to the power t/h: one halving per half-life.
We know the start, 80 g, and the end, 30 g.
t is the unknown, up in the power.
2 Get the power alone
To remove times 80, we divide by 80.
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.
80 divided by 80 is 1.
So just the power is left.
30 divided by 80 is 0.375.
Now we have (1/2) to the power t/5 = 0.375.
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/5, comes down in front.
5 Get t alone
Divide both sides by log (1/2), then times 5, to get t alone.
6 Use a calculator
With a calculator: log 0.375 is about −0.425969.
And log (1/2) is about −0.30103.
So t is about 7.08 years, to two decimal places.
7 Check
Let's check: put t = 7.08 back in.
With a calculator.
We get about 29.98 g.
That is 30, 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.