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Logarithms

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Logarithmic functions have an interesting history. Although John Napier is credited for popularizing logs, there are earlier developments that include another mathematician who developed logs earlier and independently. Furthermore, logarithms became especially important for allowing difficult calculations prior to the development of inexpensive electronic calculators. Using the Library and other resources, please research and discuss the following:

* the interpretation of natural logs and common logs
* how logarithms make scientific calculations easier
* specific applications of logarithms in science and engineering

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Logarithmic functions have an interesting history. Although John Napier is credited for popularizing logs, there are earlier developments that include another mathematician who developed logs earlier and independently. Furthermore, logarithms became especially important for allowing difficult calculations prior to the development of inexpensive electronic calculators. Using the Library and other resources, please research and discuss the following:

* the interpretation of natural logs and common logs
* how logarithms make scientific calculations easier
* specific applications of logarithms in science and engineering

Exponential functions arise when a quantity grows or shrinks by the same factor in equal time intervals (eg compound interest, population growth, radioactive decay are all examples of this). Logarithms (which are the inverse functions to exponentials) arise when we want to determine when an exponential function will reach a given value, and also in many applications involving sense perception (eg musical pitch, loudness of sounds, brightness of a star, or strength of an earthquake).

Just as subtraction is the inverse operation of addition, and taking a square root is the inverse operation of squaring, exponentiation and logarithms are inverse operations. Finding an antilog is the inverse operation of finding a log, so is another name for exponentiation. However, historically, this was done as a table lookup.
A logarithm is the power to which a number must be raised in order to get some other number. For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is ...

Solution Summary

Discussion of the history of logaritms and the interpretation of natural logs and common logs, how logarithms make scientific calculations easier and specific applications of logarithms in science and engineering.

Approximately 950 words plus 2 attachments.

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Logarithms explained in this solution

10 Problems
Please see the attached file for the fully formatted problems.
1. Find the value of x: .
Choose the correct answer from the following:

2. Evaluate the expression .

3. Write the equation in logarithmic form.

4. Evaluate the expression log 2 1.

5. Fill in the blank to make a true statement. To solve 7 x = 30, we can take the logarithm of each side of the equation to get log (7 x ) = log (30). The power rule for logarithms would then provide a way of moving the variable x from its position as an __________ to the position of a coefficient.

6. Find the value of x:

7. Assume that all variables are positive and multiply:

8. Assume that x, y and z are positive numbers. Use the properties of logarithms to write the expression in terms of the logarithms of x, y, and z.

9. Simplify the expression (256x4y)1/4 /(16xy3)3/4

1o. Assume that x is a positive number. Use the logarithm properties to present the expression log(x + 5) - log x as the logarithm of a single quantity.

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