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When evaluating a logarithmic function with a calculator, you may have noticed that the only options are or log, called the common logarithm , or ln , which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.
Let and
For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base Therefore,
In addition, we know that and are inverse functions. Therefore,
Combining these last two equalities, we conclude that
To prove the second property, we show that
Let and We will show that By the definition of logarithmic functions, we know that and From the previous equations, we see that
Therefore, Since exponential functions are one-to-one, we can conclude that
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Use a calculating utility to evaluate with the change-of-base formula presented earlier.
Use the second equation with and
Use the change-of-base formula and a calculating utility to evaluate
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 on the Richter scale and a second earthquake with magnitude on the Richter scale. Suppose which means the earthquake of magnitude 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 is the amplitude measured for the first earthquake and is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:
Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,
Therefore,
which implies or Since is 10 times the size of we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation
Therefore, That is, the first earthquake is 100 times more intense than the second earthquake.
How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?
To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:
Therefore, and we conclude that the earthquake in Japan was approximately times more intense than the earthquake in Haiti.
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