Example 3

The formula for a standard earthquake-body wave scale, , is given by , in which is the amplitude of the ground motion in microns meter), is the period of the wave, and is a correction constant. Determine a formula for the frequency of the earthquake wave if the frequency is defined as the reciprocal of the wave period. Rearrange the formula to find the range of values for when the range of values is [4, 5] seconds. Then, rearrange the formula to find the range of values of for when the range of values of is [6, 9], the range of values is [4, 5] seconds, and .
1. Substitute an expression for in place of in the formula.
2. Use the definition of the logarithm to rewrite the formula with an exponential term. Then, solve it for .
3. Use the relationship between the period and the frequency to calculate the range of frequency values.
4. Write the formula for the earthquake-magnitude scale so that the factors with known values are isolated from .
5. Use the result of the previous step to write the maximum and minimum values of the range of values for .
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