Step by step
The question
Question: A substance has a half-life of 6 hours.
How long until 40 mg decays to 5 mg?
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, 40 mg, and the end, 5 mg.
t is the unknown, up in the power.
2 Get the power alone
To remove times 40, we divide by 40.
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.
40 divided by 40 is 1.
So just the power is left.
5 divided by 40 is 0.125.
Now we have (1/2) to the power t/6 = 0.125.
3 Same base
0.125 is (1/2) to the power 3.
Same base, so the powers are equal.
So t/6 = 3.
To remove over 6, we multiply by 6.
We do it on both sides, so they stay equal.
So t = 18 hours.
Careful!
3 is the number of half-lives.
Each one is 6 hours.
4 Check
Let's check: put t = 18 back in.
We get 5 mg.
That is the end amount.
It works!
Let's look back.
First: write the equation.
Then: get the power alone.
Then: same base.
Last: check.