In Warm Up, you were introduced to the notion of logarithmic scale. One of the most familiar logarithmic scales is the Richter scale. Developed in 1935 by United States geologist Charles F. Richter, it is a scientific way of measuring the strength of earthquakes. A minor quake that rates less than 2.0 on the Richter scale is known as a micro-earthquake and is not generally strong enough to worry people. A major quake might reach upwards of 8.0 on the Richter scale and is likely to cause widespread damage and casualties.
You find the magnitude of an earthquake on the Richter scale by measuring the logarithm of how much the ground moves. Because the scale is logarithmic, every increase of one whole point on the scale means a 10 times increase in ground movement. In other words, an earthquake that measures 8.0 on the Richter scale involves 10 times move movement than an earthquake that measures 7.0, and it has 100 times more movement than a quake that measures 6.0.
The following reading in your textbook explores the Richter scale in detail and uses it to show the relationship between a logarithmic function and its corresponding exponential function.
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Read pages 426-428 Example 1 in your textbook, Principles of Mathematics 12. Complete the Reflecting questions on page 428 (A & B) to investigate further the relationship between the logarithmic and exponential forms of a function. Click here to verify your answers. |
In Lesson 7A, you explored the relationship between logarithmic and exponential graphs. You can see from the previous textbook example that a close relationship also exists between logarithmic and exponential functions. This relationship can be summarized as follows.
Consider the following example. In exponential form, you know that the base, 2, when raised to the exponent, 3, produces the answer, 8.
The corresponding logarithmic form is just the same information presented in a different arrangement.
In logarithmic form, you know that 3 is the exponent to which the base 2 must be raised to produce an answer of 8.
This relationship can be used to express logarithms as exponents and vice versa. In addition, this relationship can be used to solve equations involving exponents and logarithms. For example, in Lesson 6B, you learned how to solve exponential equations graphically if the terms of the equation could not be written with a common base. Use the relationship above as an alternate method to solve this type of question. The following textbook reading demonstrates this.