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,
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
When the exponent in a power of 10 increases by 1, say from
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