## Applications using Logarithmic Equations

### Using Logarithmic equations to solve problems

**Example #1 (Sound Intensity): The relationship between the number of decibels B and the intensity of a sound I in watts per centimeter squared is given by:**

Determine the intensity of a sound I if it registers 125 decibels on a decibel meter.

In examples #2 and 3, compare the intensities of the two earthquakes.

2.

Location |
Date |
Magnitude |

San Francisco, CA |
4/18/1906 |
8.3 |

Napa, CA |
9/3/2000 |
4.9 |

*Formula: R = log*_{10}I, where I is the intensity of the shockwave and R is the magnitude of an earthquake.

3.

Location |
Date |
Magnitude |

El Salvador |
2/13/2001 |
6.5 |

Columbia |
1/25/1999 |
5.7 |

**Example #4 (Human Memory Model): The average score A for a group of students who took a test t months after the completion of a course is given by the human memory model:**

A = 80 - log_{10}(t + 1)^{12}

How long after completing the course will the average score fall to A = 72?

**Example #5 (Newton's Law of Cooling): You place a tray of water at 60oF in a freezer that is set at 60°F. The water cools according to Newton's Law of Cooling:**

a) The water freezes in 4 hours. What is the constant k? (Hint: Water freezes at 32°F.)

b) You lower the temperature in the freezer to -10°F. At this temperature, how long will it take for the ice cubes to form?

c) The initial temperature of the water is 50°F. The freezer temperature is 0°F. How long will it take for the ice cubes to form?