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Chapter 1 | Functions and Graphs
know that b u = a , a v = x , and b w = x . From the previous equations, we see that b uv = ( b u ) v = a v = x = b w . Therefore, b uv = b w . Since exponential functions are one-to-one, we can conclude that u · v = w . □ Example 1.38 Changing Bases
Use a calculating utility to evaluate log 3 7 with the change-of-base formula presented earlier.
Solution Use the second equation with a =3 and e =3: log 3 7= ln7 ln3 ≈1.77124.
Use the change-of-base formula and a calculating utility to evaluate log 4 6.
1.32
Example 1.39 Chapter Opener: The Richter Scale for Earthquakes
Figure 1.48 (credit: modification of work by Robb Hannawacker, NPS)
In 1935, Charles Richter developed a scale (now known as the Richter scale ) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude R 1 on the Richter scale and a second earthquake with magnitude R 2 on the Richter scale. Suppose R 1 > R 2 , which means the earthquake of magnitude R 1 is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If A 1 is the amplitude measured for the first earthquake and A 2 is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation: R 1 − R 2 = log 10 ⎛ ⎝ A 1 A 2 ⎞ ⎠ . Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,
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