Math · College algebra · Worked example
Solve a half-life problem
A 400 mg dose of a drug has a half-life of 6.0 hours in the body. How much remains after 24 hours, and when does 30 mg remain?
Write the model
The amount halves every 6 hours, so after t hours there have been t/6 halvings.
Evaluate after 24 hours
24 hours is four half-lives: 400 → 200 → 100 → 50 → 25.
Set up the time equation
Set the model equal to 30 and divide both sides by 400.
Rewrite in logarithmic form
The exponent t/6 is the logarithm base 1/2 of 0.075. By change of base, log_(1/2) 0.075 = ln 0.075/ln 0.5 ≈ 3.737.
Solve and check
t ≈ 6 × 3.737 ≈ 22.4 hours. That is a little less than four half-lives, 24 hours, which fits, because 30 mg is a little more than the 25 mg left then.
Result
25 mg remains after 24 hours, and 30 mg remains after about 22.4 hours.
Your turn
At 7% continuous growth, how long does an investment take to double?
Show the answer and explanation
About 9.90 years.
Solve e^(0.07t) = 2: 0.07t = ln 2, so t = ln 2/0.07 ≈ 9.90 years. The rule of 70 gives 70/7 = 10.
Keep exploring
Open the graph and follow the curve: every 6 hours the height halves, and it crosses y = 30 at t ≈ 22.4.
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