4/5/2024 0 Comments Decibels logarithmic scaleThis not only has to do with our perception, but also how we instinctively think about numbers. That logarithmic scales often come first suggests that they are, in a sense, intuitive. Most other logarithmic scales have a similar story. In the 19th century A.D., English astronomer Norman Robert Pogson discovered that magnitude is the logarithm of the amount of starlight that hits a detector. The brightest stars in the night sky were said to be of first magnitude (m = 1), whereas the faintest were of sixth magnitude (m = 6). A good example is star brightness, which was introduced by Hipparchus, a second-century B.C. This is because a logarithmic scale is often invented first as a characterization technique without a deep understanding of the measurable phenomena behind that characterization. The table shows that the numbers relating various linear and logarithmic systems vary widely. Decibels are said to “progress arithmetically” or “vary on a logarithmic scale” because they change proportionally with the logarithm of some other measurement in this case the power of the sound wave, which “progresses geometrically” or “varies on a linear scale.” Likewise, +20 dB requires 100 times the power and +30 dB requires 1,000 times. Take sound intensity for example: To increase a speaker’s volume by 10 decibels (dB), it is necessary to supply it with 10 times the power. Logarithmic scales in scienceīecause logarithms relate multiplicative changes to incremental changes, logarithmic scales pop up in a surprising number of scientific and everyday phenomena. From our first example above, log 2(64) may be entered into a calculator as “log(64)/log(2)” or “ln(64)/ln(2)” either will give the desired answer of 6. To do a logarithm in a base other than 10 or e, we employ a property intrinsic to logarithms. Arising naturally out of the development of logarithms and calculus, it is known both as Napier’s Constant and Euler’s Number, after Leonhard Euler (1707 to 1783), a Swiss mathematician who advanced the topic a century later. The number e, which equals about 2.71828, is an irrational number (like pi) with a non-repeating string of decimals stretching to infinity. Most scientific calculators only calculate logarithms in base 10, written as log(x) for common logarithm and base e, written as ln(x) for natural logarithm (the reason why the letters l and n are backwards is lost to history). (Aside, this is less than half of the 30 C dilutions common in homeopathy, which shows why the practice is irreconcilable with modern chemistry.) Logarithms on a scientific calculator Thus, after 11 C dilutions, there will only be one molecule of the original alcohol left. Understanding that 1 ml of pure alcohol has roughly 10 22 (a one followed by 22 zeroes) molecules, how many C dilutions will it take until all but one molecule is replaced by water? Mathematically speaking, 1/100 (the base) multiplied by itself a certain number of times is 1/10 22, so how many multiplications are necessary? This question is written as: Sometimes this is referred to as a “C dilution” from Roman numeral for a hundred. When you take 1 milliliter of a liquid, add 99 ml of water, mix the solution, and then take a 1-ml sample, 99 out of every 100 molecules from the original liquid is replaced by water molecules, meaning only 1/100 of the molecules from the original liquid are left. Consequently, the base-2 logarithm of 64 is 6, so log 2(64) = 6. This means if we fold a piece of paper in half six times, it will have 64 layers.
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