The half life of a substance is the time it takes for 1/2 of the substance to decay, metabolize, or be used up. For example, if 1/2 of a drug is metabolized in 3 hours, after 6 hours 1/4 of the drug is left and 3/4 of the drug has been used:
After 9 hours, 1/8 of the drug is left: The half life formula is an exponential equation: where D is the remaining substance after time t, D_{0} is the initial amount of the substance, h is the half life of the substance, and t is the elapsed time.

Click on the blue points on the sliders in manipulative 1 and drag them to change the figure. The red points mark when 1/2, 1/4, and 1/8 of the substance has decayed. 
The half life of a radioactive substance is 3 hours. The initial amount is 3 grams. How long before only 0.6 grams is left?
Step  Equation  Description 

1  Start with the halflife formula.  
2  Fill the values into the formula. The initial amount D_{0} = 3. The halflife h = 3. The amount left after 3 hours D = 0.6. Solve for t.  
3  Use the multiplication property of equality to multiply both sides of the equation by 1/3.  
4  Use the logarithm to convert the equation from exponential form to logarithmic form. The definition of a logarithm is log_{a}b = c if and only if a^{c} = b. In this case a = 1/2, b = 0.2 and c = t/3.  
5  Use the multiplication property of equality to multiply both sides of the equation by 3.  
6  Use the change of base formula to convert the logarithm to base 10. The change of base formula is . In this case, a = 1/2, x = 0.2 and b = 10.  
7  Substitute the values of the logarithms into the equation. log_{10}0.2 ≈ 0.69897. log_{10}0.5 ≈ 0.30103.  
8  t ≈ 6.9658  Calculate the approximate value of t. It will take about 7 hours for only 0.6 grams to be left. 
Table 2: Half life example. 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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