Transcription
John Napier was a Scottish scientist best known for discovering a way to make complicated calculations easier. His development of logarithms allowed multiplication and division operations to be converted into additions and subtractions.
Napier introduced the logarithm in 1614. Napier created a list by calculating the logarithm of many numbers. After his death, other scientists extended the list. By 1624, the list included the logarithms of all the numbers from one to a hundred thousand. Using it, people could quickly find the logarithms of two or more numbers and convert multiplication operations into additions.
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For example, suppose you bought 14 apples from a market and each apple costs 12 cents. The amount you'd have to pay is 14 times 12. Using Napier's method, first we need to find the logarithm of 14 from the logarithm list. It turns out to be 1.146128. Then we need the logarithm, or log, 12. It's 1.079181. We add these to get 2.225309. Finally, we'll look at which number from the list corresponds to this log and find our answer: 168.
In this simple case, it's true that it's quicker just to do the multiplication. But the advantage of logarithms becomes clear when we start dealing with more numbers or bigger numbers.
In mathematics, logarithms is the process used to find exponents. For example, what's the power of 2 that gives 8? The answer is three. So, the logarithm of eight to base two is three. A common base to use is ten, in which case we have the following: log to base 10 of 100 is 2 because 100 is 10 squared. Log 10,000 is 4 because 10,000 is 10 to the 4.
Often, our brains perceive changes in the environment not linearly but logarithmically. For example, when we compare two light bulbs that have intensities of 25 and 50 lumens, we don't perceive the intensity of the 50 lumen bulb to be twice as great as that of the 25 lumen bulb, but much more than that. In fact, a logarithmic increase. The same is true of our sense of taste. When we compare the same amount of water with 50 and 100 grams of dissolved salt after taking a sip of each, the water containing 100 grams of salt is perceived as being much more than twice as salty as the water containing 50 grams of salt.
Research also suggests the same is true of our perception of time. As we get older, time seems to go by much faster. A 10-year-old and a 70-year-old, for instance, perceive the passage of a one-year period very differently.
In practical applications, logs are used mostly to deal with very large or very small numbers. The brightness of the sun, for instance, is about 100,000 lumens per square meter, whereas the brightest star visible at night has a brightness of [Music] 0.0000 lumens per square meter. It's hard to grasp such big differences in values, but if we switch to a log scale, the numbers become much more manageable. The brightness of the sun becomes log 100,000, which is 5, and the brightness of Sirius, the brightest star at night, is log 0.0005, or minus 4.3.
Let's take a look at some other examples of logarithms in real science. As you know, the Richter scale is used to measure the severity of earthquakes, and it's a logarithmic scale. So, an earthquake that measures six on the Richter scale isn't twice as severe as one that measures three on the Richter scale, but one thousand times more powerful.
For a substance to be an acid, it must give positively charged hydrogen ions to its environment. In the term pH, H represents these ions, while the letter p is a mathematical function that denotes minus the logarithm. So, pH is the negative log of the hydrogen ion concentration. For example, the pH value of water is about 7, and this value is considered neutral. pH 7 means log minus 7, which indicates that the hydrogen ion concentration in water is 10 to the minus 7. As you can see, very small numbers are made easier to deal with thanks to logarithms.
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The unit of sound intensity is the decibel. Louder sounds have a higher decibel value, and the decibel scale is again logarithmic. For example, a normal conversation between friends has a sound intensity of about 60 decibels. Heavy traffic averages about 70 decibels, while a noisy restaurant would register about 80 decibels. On the decibel scale, 70 decibels is 10 times noisier than 60 decibels, and 80 decibels is 100 times noisier than 60 decibels.
Our brains and sensors tend to work along logarithmic lines, so it makes sense to use logarithmic scales for measuring many of the quantities we perceive in the everyday world.