Unit 5, Lesson 18
Applications of Logarithmic Functions
Algebra 2
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Without using a calculator, put these expressions in order, from least to greatest. Be prepared to explain your reasoning.
The pH scale is a way to measure the acidity of a liquid solution. It is based on the concentration of positive hydrogen ions in the liquid. A smaller pH indicates more hydrogen ions and higher acidity. A larger pH indicates less hydrogen ions and lower acidity.
Here is a table showing the hydrogen ion concentration (in moles per liter) and the pH of some different liquids:
| liquids | hydrogen ion concentration (moles per liter) |
pH |
|---|---|---|
| pure water | 7 | |
| coffee | 5 | |
| root beer | 4 | |
| orange juice | ||
| seawater | 8 | |
| vinegar | 2.4 |
This table shows the relationship between hydrogen ion concentrations and pH ratings (acidity) for different substances.
| substance | hydrogen ion concentration (moles per liter) |
pH |
|---|---|---|
| mild detergent | 0.0000000001 | 10 |
| toothpaste | 0.000000001 | 9 |
| baking soda | 0.00000001 | 8 |
| blood | 0.0000001 | 7 |
| milk | 0.000001 | 6 |
| banana | 0.00001 | 5 |
| tomato | 0.0001 | 4 |
| apple | 0.001 | 3 |
| lemon | 0.01 | 2 |
Here is a table showing the Richter ratings for displacements recorded by a seismograph 100 km from the epicenter of an earthquake.
| seismograph displacement (meters) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Richter rating | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Compare an earthquake rated with a magnitude of 5 on the Richter scale and that rated with a 6. How do their displacements compare? What about an earthquake with a magnitude rated with a 2 and that rated with a 3?
Logarithms are helpful in a variety of real-world contexts. Let’s look at an example in chemistry.
The acidity of a substance is measured by the concentration of positive hydrogen ions, , in moles per liter. If the concentration is , then the acidity rating, or pH rating, is . For example, grapefruit juice has a hydrogen ion concentration of about mole per liter, so its acidity rating is about 3. The concentration of hydrogen ions in lemon juice is mole per liter, so its acidity or pH rating is 2.
We can see that the pH rating is -1 times the exponent in the expression representing the hydrogen ion concentration. Because the exponent in a power of 10 can be expressed in terms of the base-10 logarithm, the pH rating can be expressed as , or simply .
When the exponent in a power of 10 increases by 1, say from to , the quantity changes by a factor of 10. This means that lemon juice has 10 times the hydrogen ion concentration of grapefruit juice and the pH is 1 less. Waterhas a pH rating of 7 which is 4 greater than the pH of grapefruit juice. Water’s pH of 7 means that it has mole of hydrogen ions per liter which is of the hydrogen-ion concentration of grapefruit juice.
Another example of logarithm use is the Richter scale, which measures the strength of an earthquake in terms of the displacement of the needle on a seismograph. A displacement of 1 micrometer, one millionth of a meter, measures 1 on the Richter scale. Each time the displacement increases by a factor of 10, the Richter scale measure increases by 1. So a displacement of 10 micrometers measures 2 on the Ricther scale, and a displacement of 1,000 micrometers (1 mm) measures 4 on the Richter scale.
If the seismograph displacement is meters, the Richter rating of the earthquake can be expressed as . We can check that when (a displacement of 1 micrometer), the Richter rating is 1. And when the displacement increases by a factor of 10, the exponent of increases by 1, so the Richter rating of the earthquake increases by 1.