## Solving Applied Problems Using Exponential and Logarithmic Equations

Now, we will solve applied problems that involve half-life and the radioactive decay of chemical elements.

### Solving Applied Problems Using Exponential and Logarithmic Equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.

One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. Table 1 lists the half-life for several of the more common radioactive substances.

Substance | Use | Half-life |
---|---|---|

gallium-67 | nuclear medicine | 80 hours |

cobalt-60 | manufacturing | 5.3 years |

technetium-99m | nuclear medicine | 6 hours |

americium-241 | construction | 432 years |

carbon-14 | archeological dating | 5,715 years |

uranium-235 | atomic power | 703,800,000 years |

Table 1

where

- is the amount initially present
- is the half-life of the substance
- is the time period over which the substance is studied
- is the amount of the substance present after time

##### Example 13

*Solution*

**Analysis**

##### Try It #13

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-6-exponential-and-logarithmic-equations

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