Logarithm tables (log tables) were used to perform multiplications
prior to the advent of calculators and slide rules.
The tables below contain the logarithms base 10 of the numbers
1.000 to 1.099 and from 1.00 to
9.99, and since the logarithm function is approximately linear
beyond 1, we interpolate between values in the
table for those numbers not in the table.
A number such as 345, outside the range of the table, is first
written as 3.45 x 102
Note that log(3.45 x 102) = log(3.45) + log(102) = log(3.45) + 2
The value from the table for log(3.45)
is 0.5378, and ignoring the decimal point,
5378 is called the mantissa of
the logarithm and the number 2 is
called the characteristic of
the logarithm.
The mantissa 5378 is the same for the logarithms of 3450, 34.5,
3.45, 0.345, 0.0345, etc. Only their
characteristics differ: 3, 1, 0, -1, -2, etc.
Reading the tables to find the logarithm of a number:
First write the number in proper scientific
notation.
Unfortunately the terminology mantissa used for scientific
notation is not quite the same
as that used for logarithms. The mantissa, when speaking
logarithms, is the logarithm of the mantissa of the
number written in scientific notation. Try not to be confused
about this.
We have a number written like Ax10B. It should be
the case that 1 ≤A <10.
Next
find
the row that matches closest to A (but not greater than A.)
For example if A is 3.05 got to
row 3.0; if A is 1.05 go to row 1.05; if A is 4.65 go to row 4.6.
The values along the top of the
table get appended to the value of the row you've chosen.
For instance take A = 1.022.
The row for this number is the 3rd
row labeled 1.02. The column labeled
with 2
at the top of the table supplies the
last 2 in 1.022.
In other words that 2
is appended
to the row value 1.02 to get
1.022. Now if you look at the intersection of this column and row
you will see 0.00945. As an
exercise find the log(3.45) that I referenced earlier. (row 3.4
column 5)
In many cases there are two
different ways to find the logarithm of a number from this table as
the
following examples will explain.
Log Table 1.00 - 5.09
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| 1.00 |
0.00000 |
0.00043 |
0.00087 |
0.00130 |
0.00173 |
0.00217 |
0.00260 |
0.00303 |
0.00346 |
0.00389 |
| 1.01 |
0.00432 |
0.00475 |
0.00518 |
0.00561 |
0.00604 |
0.00647 |
0.00689 |
0.00732 |
0.00775 |
0.00817 |
| 1.02 |
0.00860 |
0.00903 |
0.00945 |
0.00988 |
0.01030 |
0.01072 |
0.01115 |
0.01157 |
0.01199 |
0.01242 |
| 1.03 |
0.01284 |
0.01326 |
0.01368 |
0.01410 |
0.01452 |
0.01494 |
0.01536 |
0.01578 |
0.01620 |
0.01662 |
| 1.04 |
0.01703 |
0.01745 |
0.01787 |
0.01828 |
0.01870 |
0.01912 |
0.01953 |
0.01995 |
0.02036 |
0.02078 |
| 1.05 |
0.02119 |
0.02160 |
0.02202 |
0.02243 |
0.02284 |
0.02325 |
0.02366 |
0.02407 |
0.02449 |
0.02490 |
| 1.06 |
0.02531 |
0.02572 |
0.02612 |
0.02653 |
0.02694 |
0.02735 |
0.02776 |
0.02816 |
0.02857 |
0.02898 |
| 1.07 |
0.02938 |
0.02979 |
0.03019 |
0.03060 |
0.03100 |
0.03141 |
0.03181 |
0.03222 |
0.03262 |
0.03302 |
| 1.08 |
0.03342 |
0.03383 |
0.03423 |
0.03463 |
0.03503 |
0.03543 |
0.03583 |
0.03623 |
0.03663 |
0.03703 |
| 1.09 |
0.03743 |
0.03782 |
0.03822 |
0.03862 |
0.03902 |
0.03941 |
0.03981 |
0.04021 |
0.04060 |
0.04100 |
| 1.0 |
0.0000 |
0.0043 |
0.0086 |
0.0128 |
0.0170 |
0.0212 |
0.0253 |
0.0294 |
0.0334 |
0.0374 |
| 1.1 |
0.0414 |
0.0453 |
0.0492 |
0.0531 |
0.0569 |
0.0607 |
0.0645 |
0.0682 |
0.0719 |
0.0755 |
| 1.2 |
0.0792 |
0.0828 |
0.0864 |
0.0899 |
0.0934 |
0.0969 |
0.1004 |
0.1038 |
0.1072 |
0.1106 |
| 1.3 |
0.1139 |
0.1173 |
0.1206 |
0.1239 |
0.1271 |
0.1303 |
0.1335 |
0.1367 |
0.1399 |
0.1430 |
| 1.4 |
0.1461 |
0.1492 |
0.1523 |
0.1553 |
0.1584 |
0.1614 |
0.1644 |
0.1673 |
0.1703 |
0.1732 |
| 1.5 |
0.1761 |
0.1790 |
0.1818 |
0.1847 |
0.1875 |
0.1903 |
0.1931 |
0.1959 |
0.1987 |
0.2014 |
| 1.6 |
0.2041 |
0.2068 |
0.2095 |
0.2122 |
0.2148 |
0.2175 |
0.2201 |
0.2227 |
0.2253 |
0.2279 |
| 1.7 |
0.2304 |
0.2330 |
0.2355 |
0.2380 |
0.2405 |
0.2430 |
0.2455 |
0.2480 |
0.2504 |
0.2529 |
| 1.8 |
0.2553 |
0.2577 |
0.2601 |
0.2625 |
0.2648 |
0.2672 |
0.2695 |
0.2718 |
0.2742 |
0.2765 |
| 1.9 |
0.2788 |
0.2810 |
0.2833 |
0.2856 |
0.2878 |
0.2900 |
0.2923 |
0.2945 |
0.2967 |
0.2989 |
| 2.0 |
0.3010 |
0.3032 |
0.3054 |
0.3075 |
0.3096 |
0.3118 |
0.3139 |
0.3160 |
0.3181 |
0.3201 |
| 2.1 |
0.3222 |
0.3243 |
0.3263 |
0.3284 |
0.3304 |
0.3324 |
0.3345 |
0.3365 |
0.3385 |
0.3404 |
| 2.2 |
0.3424 |
0.3444 |
0.3464 |
0.3483 |
0.3502 |
0.3522 |
0.3541 |
0.3560 |
0.3579 |
0.3598 |
| 2.3 |
0.3617 |
0.3636 |
0.3655 |
0.3674 |
0.3692 |
0.3711 |
0.3729 |
0.3747 |
0.3766 |
0.3784 |
| 2.4 |
0.3802 |
0.3820 |
0.3838 |
0.3856 |
0.3874 |
0.3892 |
0.3909 |
0.3927 |
0.3945 |
0.3962 |
| 2.5 |
0.3979 |
0.3997 |
0.4014 |
0.4031 |
0.4048 |
0.4065 |
0.4082 |
0.4099 |
0.4116 |
0.4133 |
| 2.6 |
0.4150 |
0.4166 |
0.4183 |
0.4200 |
0.4216 |
0.4232 |
0.4249 |
0.4265 |
0.4281 |
0.4298 |
| 2.7 |
0.4314 |
0.4330 |
0.4346 |
0.4362 |
0.4378 |
0.4393 |
0.4409 |
0.4425 |
0.4440 |
0.4456 |
| 2.8 |
0.4472 |
0.4487 |
0.4502 |
0.4518 |
0.4533 |
0.4548 |
0.4564 |
0.4579 |
0.4594 |
0.4609 |
| 2.9 |
0.4624 |
0.4639 |
0.4654 |
0.4669 |
0.4683 |
0.4698 |
0.4713 |
0.4728 |
0.4742 |
0.4757 |
| 3.0 |
0.4771 |
0.4786 |
0.4800 |
0.4814 |
0.4829 |
0.4843 |
0.4857 |
0.4871 |
0.4886 |
0.4900 |
| 3.1 |
0.4914 |
0.4928 |
0.4942 |
0.4955 |
0.4969 |
0.4983 |
0.4997 |
0.5011 |
0.5024 |
0.5038 |
| 3.2 |
0.5051 |
0.5065 |
0.5079 |
0.5092 |
0.5105 |
0.5119 |
0.5132 |
0.5145 |
0.5159 |
0.5172 |
| 3.3 |
0.5185 |
0.5198 |
0.5211 |
0.5224 |
0.5237 |
0.5250 |
0.5263 |
0.5276 |
0.5289 |
0.5302 |
| 3.4 |
0.5315 |
0.5328 |
0.5340 |
0.5353 |
0.5366 |
0.5378 |
0.5391 |
0.5403 |
0.5416 |
0.5428 |
| 3.5 |
0.5441 |
0.5453 |
0.5465 |
0.5478 |
0.5490 |
0.5502 |
0.5514 |
0.5527 |
0.5539 |
0.5551 |
| 3.6 |
0.5563 |
0.5575 |
0.5587 |
0.5599 |
0.5611 |
0.5623 |
0.5635 |
0.5647 |
0.5658 |
0.5670 |
| 3.7 |
0.5682 |
0.5694 |
0.5705 |
0.5717 |
0.5729 |
0.5740 |
0.5752 |
0.5763 |
0.5775 |
0.5786 |
| 3.8 |
0.5798 |
0.5809 |
0.5821 |
0.5832 |
0.5843 |
0.5855 |
0.5866 |
0.5877 |
0.5888 |
0.5899 |
| 3.9 |
0.5911 |
0.5922 |
0.5933 |
0.5944 |
0.5955 |
0.5966 |
0.5977 |
0.5988 |
0.5999 |
0.6010 |
| 4.0 |
0.6021 |
0.6031 |
0.6042 |
0.6053 |
0.6064 |
0.6075 |
0.6085 |
0.6096 |
0.6107 |
0.6117 |
| 4.1 |
0.6128 |
0.6138 |
0.6149 |
0.6160 |
0.6170 |
0.6180 |
0.6191 |
0.6201 |
0.6212 |
0.6222 |
| 4.2 |
0.6232 |
0.6243 |
0.6253 |
0.6263 |
0.6274 |
0.6284 |
0.6294 |
0.6304 |
0.6314 |
0.6325 |
| 4.3 |
0.6335 |
0.6345 |
0.6355 |
0.6365 |
0.6375 |
0.6385 |
0.6395 |
0.6405 |
0.6415 |
0.6425 |
| 4.4 |
0.6435 |
0.6444 |
0.6454 |
0.6464 |
0.6474 |
0.6484 |
0.6493 |
0.6503 |
0.6513 |
0.6522 |
| 4.5 |
0.6532 |
0.6542 |
0.6551 |
0.6561 |
0.6571 |
0.6580 |
0.6590 |
0.6599 |
0.6609 |
0.6618 |
| 4.6 |
0.6628 |
0.6637 |
0.6646 |
0.6656 |
0.6665 |
0.6675 |
0.6684 |
0.6693 |
0.6702 |
0.6712 |
| 4.7 |
0.6721 |
0.6730 |
0.6739 |
0.6749 |
0.6758 |
0.6767 |
0.6776 |
0.6785 |
0.6794 |
0.6803 |
| 4.8 |
0.6812 |
0.6821 |
0.6830 |
0.6839 |
0.6848 |
0.6857 |
0.6866 |
0.6875 |
0.6884 |
0.6893 |
| 4.9 |
0.6902 |
0.6911 |
0.6920 |
0.6928 |
0.6937 |
0.6946 |
0.6955 |
0.6964 |
0.6972 |
0.6981 |
| 5.0 |
0.6990 |
0.6998 |
0.7007 |
0.7016 |
0.7024 |
0.7033 |
0.7042 |
0.7050 |
0.7059 |
0.7067 |
Log Table 5.00 - 10.09
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| 5.0 |
0.6990 |
0.6998 |
0.7007 |
0.7016 |
0.7024 |
0.7033 |
0.7042 |
0.7050 |
0.7059 |
0.7067 |
| 5.1 |
0.7076 |
0.7084 |
0.7093 |
0.7101 |
0.7110 |
0.7118 |
0.7126 |
0.7135 |
0.7143 |
0.7152 |
| 5.2 |
0.7160 |
0.7168 |
0.7177 |
0.7185 |
0.7193 |
0.7202 |
0.7210 |
0.7218 |
0.7226 |
0.7235 |
| 5.3 |
0.7243 |
0.7251 |
0.7259 |
0.7267 |
0.7275 |
0.7284 |
0.7292 |
0.7300 |
0.7308 |
0.7316 |
| 5.4 |
0.7324 |
0.7332 |
0.7340 |
0.7348 |
0.7356 |
0.7364 |
0.7372 |
0.7380 |
0.7388 |
0.7396 |
| 5.5 |
0.7404 |
0.7412 |
0.7419 |
0.7427 |
0.7435 |
0.7443 |
0.7451 |
0.7459 |
0.7466 |
0.7474 |
| 5.6 |
0.7482 |
0.7490 |
0.7497 |
0.7505 |
0.7513 |
0.7520 |
0.7528 |
0.7536 |
0.7543 |
0.7551 |
| 5.7 |
0.7559 |
0.7566 |
0.7574 |
0.7582 |
0.7589 |
0.7597 |
0.7604 |
0.7612 |
0.7619 |
0.7627 |
| 5.8 |
0.7634 |
0.7642 |
0.7649 |
0.7657 |
0.7664 |
0.7672 |
0.7679 |
0.7686 |
0.7694 |
0.7701 |
| 5.9 |
0.7709 |
0.7716 |
0.7723 |
0.7731 |
0.7738 |
0.7745 |
0.7752 |
0.7760 |
0.7767 |
0.7774 |
| 6.0 |
0.7782 |
0.7789 |
0.7796 |
0.7803 |
0.7810 |
0.7818 |
0.7825 |
0.7832 |
0.7839 |
0.7846 |
| 6.1 |
0.7853 |
0.7860 |
0.7868 |
0.7875 |
0.7882 |
0.7889 |
0.7896 |
0.7903 |
0.7910 |
0.7917 |
| 6.2 |
0.7924 |
0.7931 |
0.7938 |
0.7945 |
0.7952 |
0.7959 |
0.7966 |
0.7973 |
0.7980 |
0.7987 |
| 6.3 |
0.7993 |
0.8000 |
0.8007 |
0.8014 |
0.8021 |
0.8028 |
0.8035 |
0.8041 |
0.8048 |
0.8055 |
| 6.4 |
0.8062 |
0.8069 |
0.8075 |
0.8082 |
0.8089 |
0.8096 |
0.8102 |
0.8109 |
0.8116 |
0.8122 |
| 6.5 |
0.8129 |
0.8136 |
0.8142 |
0.8149 |
0.8156 |
0.8162 |
0.8169 |
0.8176 |
0.8182 |
0.8189 |
| 6.6 |
0.8195 |
0.8202 |
0.8209 |
0.8215 |
0.8222 |
0.8228 |
0.8235 |
0.8241 |
0.8248 |
0.8254 |
| 6.7 |
0.8261 |
0.8267 |
0.8274 |
0.8280 |
0.8287 |
0.8293 |
0.8299 |
0.8306 |
0.8312 |
0.8319 |
| 6.8 |
0.8325 |
0.8331 |
0.8338 |
0.8344 |
0.8351 |
0.8357 |
0.8363 |
0.8370 |
0.8376 |
0.8382 |
| 6.9 |
0.8388 |
0.8395 |
0.8401 |
0.8407 |
0.8414 |
0.8420 |
0.8426 |
0.8432 |
0.8439 |
0.8445 |
| 7.0 |
0.8451 |
0.8457 |
0.8463 |
0.8470 |
0.8476 |
0.8482 |
0.8488 |
0.8494 |
0.8500 |
0.8506 |
| 7.1 |
0.8513 |
0.8519 |
0.8525 |
0.8531 |
0.8537 |
0.8543 |
0.8549 |
0.8555 |
0.8561 |
0.8567 |
| 7.2 |
0.8573 |
0.8579 |
0.8585 |
0.8591 |
0.8597 |
0.8603 |
0.8609 |
0.8615 |
0.8621 |
0.8627 |
| 7.3 |
0.8633 |
0.8639 |
0.8645 |
0.8651 |
0.8657 |
0.8663 |
0.8669 |
0.8675 |
0.8681 |
0.8686 |
| 7.4 |
0.8692 |
0.8698 |
0.8704 |
0.8710 |
0.8716 |
0.8722 |
0.8727 |
0.8733 |
0.8739 |
0.8745 |
| 7.5 |
0.8751 |
0.8756 |
0.8762 |
0.8768 |
0.8774 |
0.8779 |
0.8785 |
0.8791 |
0.8797 |
0.8802 |
| 7.6 |
0.8808 |
0.8814 |
0.8820 |
0.8825 |
0.8831 |
0.8837 |
0.8842 |
0.8848 |
0.8854 |
0.8859 |
| 7.7 |
0.8865 |
0.8871 |
0.8876 |
0.8882 |
0.8887 |
0.8893 |
0.8899 |
0.8904 |
0.8910 |
0.8915 |
| 7.8 |
0.8921 |
0.8927 |
0.8932 |
0.8938 |
0.8943 |
0.8949 |
0.8954 |
0.8960 |
0.8965 |
0.8971 |
| 7.9 |
0.8976 |
0.8982 |
0.8987 |
0.8993 |
0.8998 |
0.9004 |
0.9009 |
0.9015 |
0.9020 |
0.9025 |
| 8.0 |
0.9031 |
0.9036 |
0.9042 |
0.9047 |
0.9053 |
0.9058 |
0.9063 |
0.9069 |
0.9074 |
0.9079 |
| 8.1 |
0.9085 |
0.9090 |
0.9096 |
0.9101 |
0.9106 |
0.9112 |
0.9117 |
0.9122 |
0.9128 |
0.9133 |
| 8.2 |
0.9138 |
0.9143 |
0.9149 |
0.9154 |
0.9159 |
0.9165 |
0.9170 |
0.9175 |
0.9180 |
0.9186 |
| 8.3 |
0.9191 |
0.9196 |
0.9201 |
0.9206 |
0.9212 |
0.9217 |
0.9222 |
0.9227 |
0.9232 |
0.9238 |
| 8.4 |
0.9243 |
0.9248 |
0.9253 |
0.9258 |
0.9263 |
0.9269 |
0.9274 |
0.9279 |
0.9284 |
0.9289 |
| 8.5 |
0.9294 |
0.9299 |
0.9304 |
0.9309 |
0.9315 |
0.9320 |
0.9325 |
0.9330 |
0.9335 |
0.9340 |
| 8.6 |
0.9345 |
0.9350 |
0.9355 |
0.9360 |
0.9365 |
0.9370 |
0.9375 |
0.9380 |
0.9385 |
0.9390 |
| 8.7 |
0.9395 |
0.9400 |
0.9405 |
0.9410 |
0.9415 |
0.9420 |
0.9425 |
0.9430 |
0.9435 |
0.9440 |
| 8.8 |
0.9445 |
0.9450 |
0.9455 |
0.9460 |
0.9465 |
0.9469 |
0.9474 |
0.9479 |
0.9484 |
0.9489 |
| 8.9 |
0.9494 |
0.9499 |
0.9504 |
0.9509 |
0.9513 |
0.9518 |
0.9523 |
0.9528 |
0.9533 |
0.9538 |
| 9.0 |
0.9542 |
0.9547 |
0.9552 |
0.9557 |
0.9562 |
0.9566 |
0.9571 |
0.9576 |
0.9581 |
0.9586 |
| 9.1 |
0.9590 |
0.9595 |
0.9600 |
0.9605 |
0.9609 |
0.9614 |
0.9619 |
0.9624 |
0.9628 |
0.9633 |
| 9.2 |
0.9638 |
0.9643 |
0.9647 |
0.9652 |
0.9657 |
0.9661 |
0.9666 |
0.9671 |
0.9675 |
0.9680 |
| 9.3 |
0.9685 |
0.9689 |
0.9694 |
0.9699 |
0.9703 |
0.9708 |
0.9713 |
0.9717 |
0.9722 |
0.9727 |
| 9.4 |
0.9731 |
0.9736 |
0.9741 |
0.9745 |
0.9750 |
0.9754 |
0.9759 |
0.9763 |
0.9768 |
0.9773 |
| 9.5 |
0.9777 |
0.9782 |
0.9786 |
0.9791 |
0.9795 |
0.9800 |
0.9805 |
0.9809 |
0.9814 |
0.9818 |
| 9.6 |
0.9823 |
0.9827 |
0.9832 |
0.9836 |
0.9841 |
0.9845 |
0.9850 |
0.9854 |
0.9859 |
0.9863 |
| 9.7 |
0.9868 |
0.9872 |
0.9877 |
0.9881 |
0.9886 |
0.9890 |
0.9894 |
0.9899 |
0.9903 |
0.9908 |
| 9.8 |
0.9912 |
0.9917 |
0.9921 |
0.9926 |
0.9930 |
0.9934 |
0.9939 |
0.9943 |
0.9948 |
0.9952 |
| 9.9 |
0.9956 |
0.9961 |
0.9965 |
0.9969 |
0.9974 |
0.9978 |
0.9983 |
0.9987 |
0.9991 |
0.9996 |
| 10.0 |
1.0000 |
1.0004 |
1.0009 |
1.0013 |
1.0017 |
1.0022 |
1.0026 |
1.0030 |
1.0035 |
1.0039 |
When working with these tables oftentimes the decimal points can
be ignored. I will use them at first, then
later ignore them. Once you get used to using these tables you can
choose to do what you want with the decimal points.
However, the logarithm for any given number includes the decimal
point.
|
Examples
|
|
|
1)
|
Find log(1.04)
|
|
|
|
|
Start at the left column and find
the row for 1.0.
Next find the column for 4.
These two intersect at the
logarithm for 1.04.
From the table we read this to be
0.0170.
Alternatively, find 1.04 in the left column for the row then
use column 0. This gives the logarithm for 1.040. The number there
is the same (to three significant digits,) 0.01703.
|
|
2)
|
Find log (1.058)
|
|
|
|
|
Here we start at the row 1.05. That row intersects the column
whose value is 8 at the entry 0.02449.
0.02449 is the
value of log(1.058).
Now, if instead, you started at the row for 1.0 then moved to
the column for 5 that logarithm is 0.0212. This value is very
different and is due to the missing trailing 8. Notice, however,
that the logarithm for 1.058 lies somewhere between columns 5 and
6. In fact its 0.8 away form column 5. The difference between the
values in these these columns for 1.05 and 1.06 is 0.0253 –
0.0212 = 0.0410. 0.8x(.0410) = .00328. Now add this to the
smaller column value: 0.0212 + .00328 = 0.02448. We've
just interpolated between values to arrive at the
same value ±0.0001
(but the same to three significant digits) 0.0245.
|
|
3)
|
Find log(6.74)
|
|
|
|
|
This example proceeds as usual. Start at the left column and
find the row for 6.7.
Now find the column for 4 and find their intersection for the
value
0.8287.
|
|
4)
|
Find log(6.225)
|
|
|
|
|
Proceeding as before start at the left column and find the row
6.2. Now find the column 2. Now stop. This column intersects with
the row 6.2 to give the logarithm of 6.22. We need the logarithm
of 6.225.
We proceed as in example 2. 6.225 lies half way between 6.22
and 6.23. The mantissa for 6.225 must lie between the mantissas
for 6.22 and 6.23, that is, the values 0.7938 and 0.7945.
log(6.22) is 0.7938. log(6.23) = 0.7945.
6.225 is half way between 6.22 and 6.23
So, the difference in the logarithms for 6.23 and 6.22 is: 7945
– 7938 = 7 and half of 7 is 3.5.
Add 3.5 to 7938 to get 79415. So the logarithm is 0.7942.
(If you use a calculator using this procedure you will arrive
at 0.794139... and hence the value 0.7941 shown on the table. The
point of this is the last digit is caused by round off errors not
using the same number of digits.)
|
|
5)
|
Find the number whose log is 0.8344
|
|
|
|
|
Find the value in the table.
We are in luck, it exists at row 6.8 and column 3.
So immediately we have 6.8 + 0.03 = 6.83.
|
|
6)
|
Find the number whose logarithm is 0.2462.
|
|
|
|
|
This logarithm is not immediately in the table, but the values
0.2445 and 0.2480 are. They both lie in the row (and they must)
1.7. And these values append 6 and seven to 1.7 which means our
number must be between 1.76 and 1.77.
We interpolate between 2445 and 2480.
(Notice, now I am not using the decimal point.)
Here's our numbers
2455......2462......2480
This range is 2480 – 2455 = 25.
and 2462 – 2455 =
7
So the decimal we're looking for is
7/25 = 0.28
Our number becomes 1.76 + 0.0028 =
1.7628 (in 3
significant digits, this is 1.76)
|
|
|
|
|
|
7)
|
Multiply 367 and 4.2 using the log tables.
|
|
|
|
367 is 3.67 x 102 so its characteristic is 2
(the exponent of 10), we need to remember this. Likewise, 4.2 is
4.2x100, so its characteristic is 0.)
From the table:
log(3.67) is 5647
log(4.2) is 6232
now add:
5647 + 6232 = 11879
11879 is 1879 with characteristic 1 (The leftmost digit in
front of the remaining last four digits which is the mantissa. So
the characteristic of the answer will be 2 + 0 + 1 = 3.)
From the table 1879 is the logarithm of a number between 1.54
and 1.55, whose logarithms are 1875 and 1903.
The range is: 1875 .... 1879 .... 1903
1879 is 4/28 = 0.143 of this range,
so our number is (1.54 + 0.00143) x 103 = 1514.3
( 1879 – 1875 = 4
1903 – 1875 = 28
so we get 4/28 above)
|
|
|
|
|
|
|
8)
|
Multiply 32 by 0.025 using the log tables.
|
|
|
|
32 is 3.2 x 10 characteristic 1
0.025 is 2.5 x 10-2 characteristic -2
From the log tables:
log(3.2) = 5052 (1)
log(2.5) = 3979 (-2)
Add:
9031 (-1)
from the table: we find 9031 in the table and immediately have
the number 8.0 (with characteristic -1 in our calculations) so
the answer is 8.0 x 10-1 = 0.8
|
|
|
|
|
|
|
9)
|
Divide 32981 by 2435 using the log tables.
|
|
|
|
Let's do this by using 3 significant digits only.
3.30 x 104/
2.44 x 103
Remember we subtract logarithms (and therefor characteristics)
with division.
From the table we have log(3.3) – log(2.44) with
characteristic (4 – 3 = 1)
5185 – 3875 = 1310
From the table this logarithm lies between the mantissas for
1.35 and 1.36. Mentally I calculated the difference and saw that
this value lied slightly less than half way, so let's call it 0.4.
1.354 x 10 = 13.54 (and if you use a calculator the
answer is 13.54456)
Not bad!
If you carry this out as an exercise using interpolation along
the way you'll arrive at the same answer.
This exercise demonstrates why in most cases carrying three
significant digits is enough. Those of us who actually used log
tables (and slide rules) were quite good at arithmetic, namely
addition and subtraction and ratios. With the advent of the
calculator we found that we could carry out the multiplications
faster on a slide rule than they could be keyed into the
calculator. To be fair the speed of the slide rule was offset by
the number of verifications by other engineers to ensure the
calculations were correct, to three significant digits.
|
|
Now onto slide rules.
Here's a picture of a slide rule. I apologize for the scan, but
you'll get the idea.
|
|
|
|
|
|
You'll notice that most of the scales L, D, C, CI, T, S, B, A,
and K on this slide rule are not linear, but are logarithmic
scales. If you have two normal (linear) rulers and placed the
starting edge of one above the 2 inch mark of the other, then
looked to see where the 3 inch mark of the top ruler lied on the
bottom ruler you would see that on the bottom ruler that mark
would be 5 inches. Since these rulers use linear scales their
combinations would be addition.
Now if these scales were logarithmic we know that adding
logarithms means multiplying numbers. If we create a
logarithmic scales then combining them as we did with the linear
rulers means we're actually multiplying the corresponding labels
on the scales.
The next image shows the multiplication of 2 and 3. First you
would move the slide so that 1 on the C scale is directly above
the 2 on the D scale. Then you'd pinch the rule to prevent the
slide from moving with one hand and move the
cursor down the C sale until the hairline of the cursor would
lie directly over the 3 on the C scale. Then you'd follow the
hairline down to the D scale to read the result, 6.
Division was done just opposite to multiplication. For 6/3 you
would move the 3 on the C scale over the top of the 6 on the D
scale and pinch the ruler then move the cursor back to the 1 on
the D scale. Then follow the hairline down to the C scale and read
the result 2.
|
|
|
|
|
|
|
|
Scales A and B are the squares of the values on the C and D
scales.
Scale K gives the cubes of the values on scale D.
Scale L gives the mantissa of the number directly above it.
Scale CI (or sometimes R) give the reciprocals of the values on
scale D.
The S and T scales provide the sine and tangent values for
small angles.
|
|
|
|
|
|
|
This gives you an idea how calculations were done prior to
calculators. Your results not only depended on how accurately you
could read the slide rule but also how accurate you were with
basic arithmetic, that is addition and subtraction. I suspect that
since the calculator a higher degree of precision and accuracy has
been achieved, but the ability to do basic arithmetic by hand let
alone in one's head, has diminished dramatically.
|
This is another FREE TEMPLATE
PRINTABLE presented to you from the
Template section of
K12math.com