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History of Logarithms:

When were they invented?

John Napier invented logarithms in 1614, when he wrote a book called “A Description of the Wonderful Canon of Logarithms.”

Who invented them?

The word “logarithm” was made up by John Napier, a Scottish mathematician. The word “logarithm” is derived from the greek words logos and arithmos, which when put together, they mean “ratio-number.”

What country?

John Napier, the man who discovered logarithms, was from Scotland.

How were they developed/discovered?

Napier was interested in astronomy, so he spent much time dealing with large calculations and numbers, so he wanted to come up with a way to shorten the process. He called this shortened process “logarithms.”

Origins?

The concept of the logarithm originated from a desire to shorten the process of long, complicated number calculations.

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Calculating pH:

The logarithmic equation pH = -log10[H⁺] is used to calculate pH levels in substances. This equation calculates the concentration of hydrogen ions in a solution to figure out how acidic or basic a substance is. This equation is used to measure acidity in any substance, like lemon juice, or milk. It is extremely important to calculate pH in pharmaceuticals to make sure that medicines are not too acidic. It is also important to calculate pH when trying to determine whether or not a substance is dangerous.

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Measuring sound decibels:

Through using the logarithmic equation d = 10log10 I/I0, we can determine how loud a sound is. This is important because some sounds that are too loud can cause damage to the human ear. The loudness of a sound is measured in decibels. Decibels are measured by giving an intensity to a threshold sound, or the initial soft sound. The sound to be measured is given an intensity, which is calculated through using the equation previously mentioned. Without this logarithmic equation, it would be really hard to measure sound intensity since the human ear can handle such a large range of intensities. In real life, measuring the intensity of a sound is important for mixing audio.

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Calculating annual compound interest:

The equation for compound interest is A = Pe^rt. Through using this equation, we can calculate how long it will take for a bank deposit with an interest rate to reach a specific amount. In this equation, A is the final amount of money in the account after t, which is the number of years. The original amount of money is P, and the interest rate is r. Through using this equation, it is simple to calculate how much money you will have after a set amount of years with a set interest rate.

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Calculating population growth:

A populations size can be calculated through using the equation P(t) = P(0)e^kt. This equation is commonly used when a population has a constant relative growth rate. The time is relative to whatever population is being calculated, for example, when calculating population growth of bacteria, the time would be measured in days, and for humans, years. K is the constant growth rate, P(t) is population after units of time, and P(0) is the beginning population.

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Carbon dating:

This is a technique scientists use to date fossils. The logarithmic equation used for this process is much like the population growth equation. The equation used for this process is C(t) = C(0)e^-kt. The k is negative in this equation however because it is measuring decay, not growth. In real life, this equation is used not only for dating fossils, but also for determining half-lives of substances, which is helpful when performing scientific experiments using elements.

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Measuring earthquakes:

Through using the concept of logarithms, Charles Richter invented the Richter scale. Earthquakes are measured by using this scale, and it is a base-10 logarithmic scale. For example, an earthquake that measures 5.0 on the Richter scale has an amplitude that is 10 times larger than one that measures 4.0. A logarithmic scale is required because earthquakes are capable of reaching very large magnitudes, the largest ever recorded being a 9.5. The magnitude is the ratio of the amplitude of waves measured by a seismograph over a small amplitude.

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Sources:

https://www.khanacademy.org/math/algebra2/logarithms-tutorial http://www.mathsisfun.com/algebra/logarithms.html http://math.about.com/library/weekly/blbionapier.htm http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/log_exp/realworldappslogarithm.html http://documentation.apple.com/en/finalcutpro/usermanual/index.html#chapter=52%26section=2%26tasks=true http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson http://beginnersinvest.about.com/lw/Business-Finance/Personal-finance/Calculating-Logarithms-and-Compound-Interest.htm http://www.sparknotes.com/math/precalc/exponentialandlogarithmicfunctions/section3.rhtml http://tremor.nmt.edu/faq/how.html http://en.wikipedia.org/wiki/Richter_magnitude_scale