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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Common logarithm ： ウィキペディア英語版 Common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and also as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as "standard logarithm". It is indicated by log10(''x''), or sometimes Log(''x'') with a capital ''L'' (however, this notation is ambiguous since it can also mean the complex natural logarithmic multi-valued function). On calculators it is usually "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity the ISO 80000 specification recommends that log10(''x'') should be written lg (''x'') and log''e''(''x'') should be ln (''x''). ==Uses== Before the early 1970s, handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation when calculations required greater accuracy than could be achieved with a slide rule. Use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well.〔E. R. Hedrick, (Logarithmic and Trigonometric Tables ) (Macmillan, New York, 1913).〕 See log table for the history of such tables. An important property of base-10 logarithms which makes them so useful in calculation is that the logarithm of numbers greater than one which differ by a power of ten all have the same fractional part. The fractional part is known as the mantissa.〔This use of the word ''mantissa'' stems from an older, non-numerical, meaning: a minor addition or supplement, e.g. to a text. Nowadays, the word ''mantissa'' is generally used to describe the fractional part of a floating point number on computers, though the recommended term is significand.〕 Thus log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to 4 or 5 decimal places or more, of each number in a range, e.g. 1000 to 9999. Such a range would cover all possible values of the mantissa. The integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by: :$\backslash log\_120=\backslash log\_(10^2\backslash times\; 1.2)=2+\backslash log\_1.2\backslash approx2+0.07918.$ The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2. Numbers greater than 0 and less than 1 have negative logarithms. For example, :$\backslash log\_0.012=\backslash log\_(10^\backslash times\; 1.2)=-2+\backslash log\_1.2\backslash approx-2+0.07918=-1.92082$ To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, a bar notation is used: :$\backslash log\_0.012\backslash approx-2+0.07918=\backslash bar.07918$ The bar over the characteristic indicates that it is negative whilst the mantissa remains positive. When reading a number in bar notation out loud, the symbol $\backslash bar$ is read as "bar n", so that $\backslash bar.07918$ is read as "bar 2 point 07918...". The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102: :$\backslash begin$ \text &\log_0.012\approx\bar.07918 \\ \text\;\;\log_0.85&=\log_(10^\times 8.5)=-1+\log_8.5&\approx-1+0.92942=\bar.92942\;, \\ \log_(0.012\times 0.85) &=\log_0.012+\log_0.85 &\approx\bar.07918+\bar.92942 \\ &=(-2+0.07918)+(-1+0.92942) &=-(2+1)+(0.07918+0.92942) \\ &=-3+1.00860 &=-2+0.00860\;^ * \\ &\approx\log_(10^)+\log_(1.02) &=\log_(0.01\times 1.02) \\ &=\log_(0.0102) \end * This step makes the mantissa between 0 and 1, so that its antilog (10mantissa) can be looked up. The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten: Note that the mantissa is common to all of the 5×10''i''. This holds for any positive real number $x$ because: :$\backslash log\_(x\backslash times10^i)=\backslash log\_(x)+\backslash log\_(10^i)=\backslash log\_(x)+i$. Since $i$ is always an integer the mantissa comes from $\backslash log\_(x)$ which is constant for given $x$. This allows a table of logarithms to include only one entry for each mantissa. In the example of 5×10''i'', 0.698 970 (004 336 018 ...) will be listed once indexed by 5, or 0.5, or 500 etc.. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Common logarithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.