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常用対数表

本項では以下の内容を解説しています。

  • ・常用対数表の使い方
  • ・常用対数表に関連する例題
  • ・常用対数表

【1】常用対数表の使い方

常用対数とは底が\(\large{\hspace{1pt}10\hspace{2pt}}\)の対数\(\large{\hspace{1pt}\log_{10} \rm{M}\hspace{3pt}}\)のことをいいます。

例えば\(\large{\hspace{2pt}\rm{M}= 100\hspace{3pt}}\)の常用対数は $$\large{\log_{10}100 = \log_{10}10^2 = 2 }$$ また、\(\large{\rm{M}= 1000\hspace{3pt}}\)の常用対数は $$\large{\log_{10}1000 = \log_{10}10^3 = 3 }$$ となります。

・常用対数の例

よく使われる常用対数をまとめると、以下の表のようになります。

真数 常用対数の値
\(\displaystyle \large{1}\) \(\displaystyle \large{\log_{10} 1 = 0}\)
\(\displaystyle \large{2}\) \(\displaystyle \hspace{25pt}\large{\log_{10} 2 = 0.3010}\)
\(\displaystyle \large{3}\) \(\displaystyle \hspace{25pt}\large{\log_{10} 3 = 0.4771}\)
\(\displaystyle \large{7}\) \(\displaystyle \hspace{25pt}\large{\log_{10} 7 = 0.8451}\)
\(\displaystyle \large{10}\) \(\displaystyle \hspace{5pt}\large{\log_{10} 10 = 1}\)
\(\displaystyle \large{100}\) \(\displaystyle \hspace{9pt}\large{\log_{10} 10^2 = 2}\)
\(\displaystyle \large{1000}\) \(\displaystyle \hspace{9pt}\large{\log_{10} 10^3 = 3}\)

・常用対数表とは

一般的にある正の数\(\large{\hspace{1pt}\rm{M}\hspace{2pt}}\)は $$\large{\rm{M}= \mathit{a} \times 10^\mathit{n} }$$ と表されます。(ただし\(\large{\hspace{2pt}1 \leqq a < 10\hspace{2pt}}\)、\(\large{n\hspace{2pt}}\)は整数)

ここで\(\large{\hspace{3pt}\rm{M}= \mathit{a} \times 10^\mathit{n} \hspace{3pt}}\)に常用対数をとると \begin{eqnarray} \large \log_{10} \rm{M} &\large =&\large \log_{10} (\mathit{a} \times 10^\mathit{n})\\[0.7em] \large &\large =&\large \log_{10} \mathit{a} + \log_{10}10^\mathit{n} \\[0.7em] \large &\large =&\large \color{red}{\log_{10} a}\color{black}{} + \color{blue}{n} \\[0.7em] \end{eqnarray} となります。

このとき\(\large{\hspace{3pt}1 \leqq a < 10\hspace{2pt}}\)であることから、\(\large{0 \leqq \log_{10}a < 1\hspace{2pt}}\)となります。
また、\(\large{n\hspace{2pt}}\)は値\(\large{\hspace{1pt}\rm{M}\hspace{2pt}}\)の桁数に対応する整数となります。

すなわち、任意の正の数\(\large{\hspace{1pt}\rm{M}\hspace{2pt}}\)の常用対数\(\large{\hspace{1pt}\log_{10}\rm{M}\hspace{2pt}}\)を求めたい場合は $$\large{\log_{10} \rm{M} = \color{red}{\log_{10} \mathit{a}}\color{black}{} + \color{blue}{n}}$$ の形に変形することで、真数が\(\large{\hspace{1pt}1\hspace{2pt}}\)以上\(\large{\hspace{1pt}10\hspace{2pt}}\)未満の常用対数\(\large{\hspace{1pt}\color{red}{\log_{10} \mathit{a}}\hspace{2pt}}\)と整数\(\large{\hspace{1pt}n\hspace{2pt}}\)の値から求めることができます。

真数が\(\large{\hspace{1pt}1\hspace{2pt}}\)以上\(\large{\hspace{1pt}10\hspace{2pt}}\)未満の常用対数の値をまとめた表を常用対数表といいます。

常用対数表は本項の後半に記載しています。

・常用対数表の使い方

常用対数表の使い方は、縦方向に『真数の上から二桁』、横方向に『真数の小数点第二位』の値を探し、その行と列が重なる位置の値を読みます。

例えば、\(\large{\hspace{1pt}\log_{10}2.15\hspace{2pt}}\)の場合は縦方向に『真数の上から二桁の\(\large{\hspace{1pt}2.1\hspace{2pt}}\)』、横方向に『真数の小数点第二位の\(\large{\hspace{1pt}5\hspace{2pt}}\)』を見つけて行と列が重なる位置の値を読みます。

以下の常用対数表から\(\large{\hspace{2pt}.3324\hspace{2pt}}\)とあるため常用対数\(\large{\hspace{1pt}\log_{10}2.15\hspace{2pt}}\)の値は $$\large{\large{\hspace{1pt}\log_{10}2.15\hspace{2pt}=0.3324}}$$ となります。 常用対数表の見方

・例題1

【例題1】
常用対数表を用いて以下の値を小数点\(\large{\hspace{1pt}4\hspace{2pt}}\)位まで求めよ
\(\large{\hspace{5pt}\log_{10}1.55}\)

常用対数表から縦方向に『真数の上二桁の\(\large{\hspace{1pt}1.5\hspace{2pt}}\)』、横方向に『真数の小数点第二位の\(\large{\hspace{1pt}5\hspace{2pt}}\)』を見つけ、その行と列が重なる値を読むと\(\large{\hspace{1pt}.1903\hspace{2pt}}\)となります。

したがって $$\large{\log_{10}1.55 = 0.1903}$$ と求められます。

・例題2

【例題2】
常用対数表を用いて以下の値を小数点\(\large{\hspace{1pt}4\hspace{2pt}}\)位まで求めよ
\(\large{\hspace{5pt}\log_{10}31400\hspace{8pt}}\)

\(\large{\hspace{1pt}\log_{10}31400\hspace{2pt}}\)を変形すると \begin{eqnarray} \large \log_{10} 31400 &\large =&\large \log_{10} (3.14 \times 10^4)\\[0.7em] \large &\large =&\large \log_{10} 3.14 + \log_{10}10^4 \\[0.7em] \large &\large =&\large \log_{10} 3.14 + 4 \\[0.7em] \end{eqnarray} ここで、常用対数表から\(\large{\hspace{1pt}\log_{10} 3.14\hspace{2pt}}\)の値を求めると縦方向に『真数の上二桁の\(\large{\hspace{1pt}3.1\hspace{2pt}}\)』、横方向に『真数の小数点第二位の\(\large{\hspace{1pt}4\hspace{2pt}}\)』を見つけ、その行と列が重なる値を読むと\(\large{\hspace{1pt}.4969\hspace{2pt}}\)となるため $$\large{\log_{10}3.14 = 0.4969}$$ と求められます。

したがって、 \begin{eqnarray} \large \log_{10} 31400 &\large =&\large \log_{10} 3.14 + 4\\[0.7em] \large &\large =&\large 0.4969 + 4\\[0.7em] \large &\large =&\large 4.4969 \\[0.7em] \end{eqnarray} と求められます。

・例題3

【例題3】
常用対数表を用いて以下の値を小数点\(\large{\hspace{1pt}4\hspace{2pt}}\)位まで求めよ
\(\large{\hspace{5pt}\log_{10}0.00119}\)

\(\large{\hspace{1pt}\log_{10}0.00119\hspace{2pt}}\)を変形すると \begin{eqnarray} \large \log_{10} 0.00119 &\large =&\large \log_{10} (1.19 \times 10^{-3})\\[0.7em] \large &\large =&\large \log_{10} 1.19 + \log_{10}10^{-3} \\[0.7em] \large &\large =&\large \log_{10} 1.19 -3 \\[0.7em] \end{eqnarray} ここで、常用対数表から\(\large{\hspace{1pt}\log_{10} 1.19\hspace{2pt}}\)の値を求めると縦方向に『真数の上二桁の\(\large{\hspace{1pt}1.1\hspace{2pt}}\)』、横方向に『真数の小数点第二位の\(\large{\hspace{1pt}9\hspace{2pt}}\)』を見つけ、その行と列が重なる値を読むと\(\large{\hspace{1pt}.0755\hspace{2pt}}\)となるため $$\large{\log_{10}1.19 = 0.0755}$$ と求められます。

したがって、 \begin{eqnarray} \large \log_{10} 0.00119 &\large =&\large \log_{10} 1.19 -3\\[0.7em] \large &\large =&\large 0.0755 -3\\[0.7em] \large &\large =&\large -2.9245 \\[0.7em] \end{eqnarray} と求められます。

【2】常用対数表

本章では、常用対数表を示します。

・常用対数表①

常用対数の値
0 1 2 3 4 5 6 7 8 9
1.0 .0000 .0043 .0086 .0128 .0170 .0212 .0253 .0294 .0334 .0374
1.1 .0414 .0453 .0492 .0531 .0569 .0607 .0645 .0682 .0719 .0755
1.2 .0792 .0828 .0864 .0899 .0934 .0969 .1004 .1038 .1072 .1106
1.3 .1139 .1173 .1206 .1239 .1271 .1303 .1335 .1367 .1399 .1430
1.4 .1461 .1492 .1523 .1553 .1584 .1614 .1644 .1673 .1703 .1732
1.5 .1761 .1790 .1818 .1847 .1875 .1903 .1931 .1959 .1987 .2014
1.6 .2041 .2068 .2095 .2122 .2148 .2175 .2201 .2227 .2253 .2279
1.7 .2304 .2330 .2355 .2380 .2405 .2430 .2455 .2480 .2504 .2529
1.8 .2553 .2577 .2601 .2625 .2648 .2672 .2695 .2718 .2742 .2765
1.9 .2788 .2810 .2833 .2856 .2878 .2900 .2923 .2945 .2967 .2989
2.0 .3010 .3032 .3054 .3075 .3096 .3118 .3139 .3160 .3181 .3201
2.1 .3222 .3243 .3263 .3284 .3304 .3324 .3345 .3365 .3385 .3404
2.2 .3424 .3444 .3464 .3483 .3502 .3522 .3541 .3560 .3579 .3598
2.3 .3617 .3636 .3655 .3674 .3692 .3711 .3729 .3747 .3766 .3784
2.4 .3802 .3820 .3838 .3856 .3874 .3892 .3909 .3927 .3945 .3962
2.5 .3979 .3997 .4014 .4031 .4048 .4065 .4082 .4099 .4116 .4133
2.6 .4150 .4166 .4183 .4200 .4216 .4232 .4249 .4265 .4281 .4298
2.7 .4314 .4330 .4346 .4362 .4378 .4393 .4409 .4425 .4440 .4456
2.8 .4472 .4487 .4502 .4518 .4533 .4548 .4564 .4579 .4594 .4609
2.9 .4624 .4639 .4654 .4669 .4683 .4698 .4713 .4728 .4742 .4757
3.0 .4771 .4786 .4800 .4814 .4829 .4843 .4857 .4871 .4886 .4900
3.1 .4914 .4928 .4942 .4955 .4969 .4983 .4997 .5011 .5024 .5038
3.2 .5051 .5065 .5079 .5092 .5105 .5119 .5132 .5145 .5159 .5172
3.3 .5185 .5198 .5211 .5224 .5237 .5250 .5263 .5276 .5289 .5302
3.4 .5315 .5328 .5340 .5353 .5366 .5378 .5391 .5403 .5416 .5428
3.5 .5441 .5453 .5465 .5478 .5490 .5502 .5514 .5527 .5539 .5551
3.6 .5563 .5575 .5587 .5599 .5611 .5623 .5635 .5647 .5658 .5670
3.7 .5682 .5694 .5705 .5717 .5729 .5740 .5752 .5763 .5775 .5786
3.8 .5798 .5809 .5821 .5832 .5843 .5855 .5866 .5877 .5888 .5899
3.9 .5911 .5922 .5933 .5944 .5955 .5966 .5977 .5988 .5999 .6010
4.0 .6021 .6031 .6042 .6053 .6064 .6075 .6085 .6096 .6107 .6117
4.1 .6128 .6138 .6149 .6160 .6170 .6180 .6191 .6201 .6212 .6222
4.2 .6232 .6243 .6253 .6263 .6274 .6284 .6294 .6304 .6314 .6325
4.3 .6335 .6345 .6355 .6365 .6375 .6385 .6395 .6405 .6415 .6425
4.4 .6435 .6444 .6454 .6464 .6474 .6484 .6493 .6503 .6513 .6522
4.5 .6532 .6542 .6551 .6561 .6571 .6580 .6590 .6599 .6609 .6618
4.6 .6628 .6637 .6646 .6656 .6665 .6675 .6684 .6693 .6702 .6712
4.7 .6721 .6730 .6739 .6749 .6758 .6767 .6776 .6785 .6794 .6803
4.8 .6812 .6821 .6830 .6839 .6848 .6857 .6866 .6875 .6884 .6893
4.9 .6902 .6911 .6920 .6928 .6937 .6946 .6955 .6964 .6972 .6981
5.0 .6990 .6998 .7007 .7016 .7024 .7033 .7042 .7050 .7059 .7067
5.1 .7076 .7084 .7093 .7101 .7110 .7118 .7126 .7135 .7143 .7152
5.2 .7160 .7168 .7177 .7185 .7193 .7202 .7210 .7218 .7226 .7235
5.3 .7243 .7251 .7259 .7267 .7275 .7284 .7292 .7300 .7308 .7316
5.4 .7324 .7332 .7340 .7348 .7356 .7364 .7372 .7380 .7388 .7396

・常用対数表②

常用対数の値
0 1 2 3 4 5 6 7 8 9
5.5 .7404 .7412 .7419 .7427 .7435 .7443 .7451 .7459 .7466 .7474
5.6 .7482 .7490 .7497 .7505 .7513 .7520 .7528 .7536 .7543 .7551
5.7 .7559 .7566 .7574 .7582 .7589 .7597 .7604 .7612 .7619 .7627
5.8 .7634 .7642 .7649 .7657 .7664 .7672 .7679 .7686 .7694 .7701
5.9 .7709 .7716 .7723 .7731 .7738 .7745 .7752 .7760 .7767 .7774
6.0 .7782 .7789 .7796 .7803 .7810 .7818 .7825 .7832 .7839 .7846
6.1 .7853 .7860 .7868 .7875 .7882 .7889 .7896 .7903 .7910 .7917
6.2 .7924 .7931 .7938 .7945 .7952 .7959 .7966 .7973 .7980 .7987
6.3 .7993 .8000 .8007 .8014 .8021 .8028 .8035 .8041 .8048 .8055
6.4 .8062 .8069 .8075 .8082 .8089 .8096 .8102 .8109 .8116 .8122
6.5 .8129 .8136 .8142 .8149 .8156 .8162 .8169 .8176 .8182 .8189
6.6 .8195 .8202 .8209 .8215 .8222 .8228 .8235 .8241 .8248 .8254
6.7 .8261 .8267 .8274 .8280 .8287 .8293 .8299 .8306 .8312 .8319
6.8 .8325 .8331 .8338 .8344 .8351 .8357 .8363 .8370 .8376 .8382
6.9 .8388 .8395 .8401 .8407 .8414 .8420 .8426 .8432 .8439 .8445
7.0 .8451 .8457 .8463 .8470 .8476 .8482 .8488 .8494 .8500 .8506
7.1 .8513 .8519 .8525 .8531 .8537 .8543 .8549 .8555 .8561 .8567
7.2 .8573 .8579 .8585 .8591 .8597 .8603 .8609 .8615 .8621 .8627
7.3 .8633 .8639 .8645 .8651 .8657 .8663 .8669 .8675 .8681 .8686
7.4 .8692 .8698 .8704 .8710 .8716 .8722 .8727 .8733 .8739 .8745
7.5 .8751 .8756 .8762 .8768 .8774 .8779 .8785 .8791 .8797 .8802
7.6 .8808 .8814 .8820 .8825 .8831 .8837 .8842 .8848 .8854 .8859
7.7 .8865 .8871 .8876 .8882 .8887 .8893 .8899 .8904 .8910 .8915
7.8 .8921 .8927 .8932 .8938 .8943 .8949 .8954 .8960 .8965 .8971
7.9 .8976 .8982 .8987 .8993 .8998 .9004 .9009 .9015 .9020 .9025
8.0 .9031 .9036 .9042 .9047 .9053 .9058 .9063 .9069 .9074 .9079
8.1 .9085 .9090 .9096 .9101 .9106 .9112 .9117 .9122 .9128 .9133
8.2 .9138 .9143 .9149 .9154 .9159 .9165 .9170 .9175 .9180 .9186
8.3 .9191 .9196 .9201 .9206 .9212 .9217 .9222 .9227 .9232 .9238
8.4 .9243 .9248 .9253 .9258 .9263 .9269 .9274 .9279 .9284 .9289
8.5 .9294 .9299 .9304 .9309 .9315 .9320 .9325 .9330 .9335 .9340
8.6 .9345 .9350 .9355 .9360 .9365 .9370 .9375 .9380 .9385 .9390
8.7 .9395 .9400 .9405 .9410 .9415 .9420 .9425 .9430 .9435 .9440
8.8 .9445 .9450 .9455 .9460 .9465 .9469 .9474 .9479 .9484 .9489
8.9 .9494 .9499 .9504 .9509 .9513 .9518 .9523 .9528 .9533 .9538
9.0 .9542 .9547 .9552 .9557 .9562 .9566 .9571 .9576 .9581 .9586
9.1 .9590 .9595 .9600 .9605 .9609 .9614 .9619 .9624 .9628 .9633
9.2 .9638 .9643 .9647 .9652 .9657 .9661 .9666 .9671 .9675 .9680
9.3 .9685 .9689 .9694 .9699 .9703 .9708 .9713 .9717 .9722 .9727
9.4 .9731 .9736 .9741 .9745 .9750 .9754 .9759 .9763 .9768 .9773
9.5 .9777 .9782 .9786 .9791 .9795 .9800 .9805 .9809 .9814 .9818
9.6 .9823 .9827 .9832 .9836 .9841 .9845 .9850 .9854 .9859 .9863
9.7 .9868 .9872 .9877 .9881 .9886 .9890 .9894 .9899 .9903 .9908
9.8 .9912 .9917 .9921 .9926 .9930 .9934 .9939 .9943 .9948 .9952
9.9 .9956 .9961 .9965 .9969 .9974 .9978 .9983 .9987 .9991 .9996


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