Applications of Logarithmic Functions

The measure of sound intensity (decibels), the measure of acidity in a solution (the pH factor), and the measure of the extent to which the earth vibrates during an earthquake (the Richter value) are all examples of logarithmic functions.

pH measures the concentration of the hydrogen ion in a solution. (Brackets are used in chemical terminology to stand for concentration, so the symbol [H+] is read, “the concentration of the hydrogen ion.”) pH is computed as the negative logarithm of [H+]; that is, pH = -log([H+]). A solution with a high concentration of the hydrogen ion—that is, a large [H+] value—is called an acid, and a solution with a low concentration of the hydrogen ion—that is, a low [H+] value—is called a base.

Determine the [H+] for lemon juice, which has a pH of 2.4.

pH = -log[H+] so for this problem, 2.4 = -log[H+]. Changing the equation to exponential form, we get [H+]= 10-2.4 = 0.003981.

The earth vibrates continuously but at different rates in different places. The normal intensity of the vibrations in the earth at a given location is designated by the variable I0. The Richter scale compares the intensity of vibrations during a seismic event to this base number and reports the answer as a logarithm. That is,

, where I is the intensity of the vibration during the earthquake.

The earthquake that struck Port-au-Prince, Haiti, in 2010 measured 7.0 on the Richter scale. Determine the intensity of this quake in terms of the earth's normal vibrations in Port-au-Prince.

. Written in exponential form, this becomes
, or I = 107 I0. The intensity of the vibrations during the earthquake in Port-au-Prince was 10 million times the normal vibrations of the earth at that location!

Logarithms are also used to solve exponential equations algebraically. One word of warning: Concentrate on solving the equation, and don't worry about the strange notation in the solution.

Similar to the measurement of vibrations of the earth during a seismic event, sound is measured relative to a normal value—in this case the threshold intensity of sound, I0. The number of decibels, db, is given by

.

Airport employees wear protective ear covers, because the decibel reading for planes on an airport runway is 120 db. How many times greater than the threshold of sound is the sound of a plane on a runway?

. Rewrite as an exponential equation to get
. The intensity of sound on the runway, I, is 1012 times the threshold of sound, I0.

By the time a 14-pound turkey is taken out of the refrigerator, stuffed, and put into the 325oF oven, the temperature of the turkey is approximately 40oF. The temperature of the turkey t hours after it is put into the oven is given by T(t) = 325 − 285e(-0.228x). Dinner will be ready 30 minutes after the temperature of the turkey reaches 180oF. If you put the turkey into the oven at 1:00 o'clock, when do you eat?

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