MAT 150 Module 9 – Logarithmic Functions Lesson 2 – Applications of Logarithms

Applications of logarithms Logarithmic scales are used for measurement. A logarithmic scale is often used when we are not interested in the absolute difference between two quantities, but in their size relative to each other. Since logarithmic scales measure ratios, they do not have units associated with them.

Logarithmic Scales For example, say a logarithmic scale with base 10 is used for measurement. Each increase of one point on the scale means a measurement is ten times as large. Base 10 is the most common base since we use the base 10 number system.

The Richter Scale The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale with base 10. So every increase of 1 point on the Richter scale results in an earthquake that is 10 times stronger.

The Richter Scale

Example – The Richter Scale A.What is the Richter scale measurement of an earthquake that measures 45,000 times I 0 ? B.If an earthquake measures 4.5 on the Richter scale How many times stronger is it than I 0 ? How many times stronger is it than an earthquake that measures 2.3?

Solution

Solutions B. If an earthquake measures 4.5 on the Richter scale How many times stronger is it than I 0 ? How many times stronger is it than an earthquake that measures 2.3? An earthquake measuring 4.5 on the Richter scale is ≈ 31,623 times I 0. The Richter scale difference between an earthquake measuring 4.5 and an earthquake measuring 2.3 is 2.2. So their difference in magnitude is ≈ So the 4.5 earthquake is about 158 times stronger than the 2.3 earthquake.

Logarithmic Scales Other examples of common logarithmic scales include the Decibel scale (used to measure the loudness of sounds) and the pH scale (used to measure the acidity of a solution).

Logarithmic Scales The Decibel Scale uses a similar scale to the Richter scale to measure the intensity, or loudness of a sound. D = 10log Where D is the the measurement of the loudness of the sound, I is the measured intensity of the sound, and I 0 is the minimum intensity of a sound that can be heard.

Octaves If you are a musician, you should also know that the octave is a logarithmic scale. The octave scale is a base 2 logarithmic scale. This means that for any given musical note, a note one octave higher will have a frequency twice as high, and a note one octave lower will have a frequency ½ as high.

The pH scale The pH scale is used in science to measure the acidity of solutions. This scale is another base 10 scale. The formula for the pH of a solution is pH = -log(H + ) where H + is the concentration of positively charged Hydrogen ions in the solution.

The pH scale - Example If a solution has a concentration of hydrogen ions equal to moles per liter, what is the pH of the solution?

The pH scale - Example If a solution has a concentration of hydrogen ions equal to moles per liter, what is the pH of the solution? –log( ) = 2.437

The pH scale - Example Compare the solution in the previous example, with a pH of 2.4 with a solution that has a pH of 4.5. How much more acidic is this solution than a solution with a pH of 4.5?

The pH scale – Example.4 How much more acidic is this solution than a solution with a pH of 4.5? The difference in the pH is 4.5 – 2.4 = 2.1. The difference in acidity is = The solution with a pH of 2.4 is approximately 126 times more acidic than the solution with a pH of 4.5.

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