100 000000 000434 000868 001301 001734 002166 002598 003029 003461003891 432 1 4321 4751 5181 5609 6038 6466 689-4 7321 7748 8174 428 2 8600 9026 9451 9876 010300 010724 011147 011570 011993 012415 424 3 012837 013259 013680 014100 4521 4940 5360 5779 6197 6616 420 4 7033 7451 7868 8284 8700 9116 9532 9947 020361 020775 416 5 021189 021603 022016 022428 022841 023252 023664 024075 4486 4896 412 6 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 7 9384 9789 030195 030600 031001 031403 031812 032216 032619 033021 404 8 033424 033826 4227 4628 5029 5430 5830 6230 6629 7028 400 9 7426 7825 8223 8620 9017 9414 9811 040207 010602 040998 397 110 041393 041787 042182 012576 042969 043362 043755 044148 044540 044932 393

1

5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 390

051924 052309 052694 386

234 10

5378

9218 9606 9993 050380 050766 051153 051538 3 053078 053463 053846 4230 4613 4996 5760 6142 6524 383 6905 7286 7666 8046 8426 8805 9185 9563 9942 060320 379 5 060698 061075 061452 061829 062206 062582 062958 063333 063709 4083 376 6 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 7 8186 8557 8928 9298 9668 070038 070407 070776 071145 071514 370 8 071882 072250 072617 072985 073352 3718 4085 4451 4816 5182 366 9 5547 5912 6276 6640 7004 7368 7731 8091 8457 8819 363 $120 09181 079513 079904030266 08066 080937 081347 081707 082067 082426 360

The logarithms in the preceding tables are taken in the common system, and are sufficient to illustrate the use of a larger table, such as that appended to the author's Trigonometry. In Table II. the first two figures of several successive numbers, and of the logarithms, when the same, are left to be supplied.

414. Numbers may be formed in Table II. by annexing each figure at the top of the table to the figures in the column at the left, and the decimal of the logarithm of the number formed will be in a line with the figures taken at the left, and in the column with the figure taken at the top. Thus,

415. When a number is not in the table, if its factors are there, its logarithm may be found by taking out from the table the logarithms of the factors, and adding them together (Art. 399). Thus, since

7248 = 1208 × 6, log 7248 = log 1208log 6

= 3.082067 +0.778151 = 3.860218.

416. When a number is intermediate between those in the table, its logarithm may be found by interpolation.

For, the tabulated logarithms are of the nature of a regular series, hence they may be interpolated by the formula in Art. 387. Thus,

Let it be required to find the logarithm of 11.887. Here, 11.887 is a number between 11.88 and 11.89; then log 11.89

= .000366.

1.074816 - log 11.88 1.075182

Hence, put d1 = .000366, and p .7, and we have

This process is based upon the supposition that the differences of logarithms are proportional to the differences of their corresponding numbers, which is not strictly correct, yet sufficiently exact for practical purposes.

The difference d might have been taken directly from the table, being found in a line with the logarithm of 11.88, and in column D.

417. When a given logarithm is intermediate between those found in the table, its corresponding number may be found.

For, this is obviously the converse of the last case, and, therefore depends upon a process based upon a like supposition. Thus,

Let it be required to find the number corresponding to the logarithm 1.075072.

The decimal of the given log is .075072

66

Difference,

the log next less ".074816, corresp'ng no. 11.88

256.0

Diff. betw. log 11.88 and log 11.89, 366

Whence, the required number

EXAMPLES.

1. Find the logarithm of 1095. 2. Find the logarithm of .0735.

Ans. 3.039414.

Ans. 2.866287.

=

Ans. 2.447737.

Ans. 1.245727.

1740; log 405 log 81+ log 5-log 23.

=

9. Find the logarithm of 1.0375. 10. Find the logarithm of 104.857.

Ans. 0.015988.

Ans. 2.020597.

11. Find the number whose logarithm is 0.025306.

Ans. 1.06.

12. Find the number whose logarithm is 3.010724. 13. Find the number whose logarithm is 0.009663. Ans. 1.0225.

14. Find the number whose logarithm is 3.009264.

Ans. 1021.56.

EXPONENTIAL EQUATIONS.

418. An EXPONENTIAL QUANTITY is one that has an unknown quantity for an exponent; as,

1

a2, a*, x*, etc.

419. An EXPONENTIAL EQUATION is one containing an exponential quantity, as

ax = b, x* = a, etc.

Equations of this kind may be readily solved by logarithms.

420. To solve an exponential equation in the form of

ax= b,

take the logarithm of each member, and we have

1. Find the value of x in the equation 2o = 1024.

2. Find the value of x in the equation 3* = 15. Ans. 2.465.

3. Find the value of x in the equation 5* = 100. Ans. 2.861.

4. Find the value of x in the equation ab* = c.

5. Find the value of x in the equation 3* = 9.

Here, raising both members to the xth power, we have 9o, or 81 = 9; and log 81 = x log 9 ;

34 =

6. Find the value of x in the equation 6* = 3.

Ans. 3.911.

7. Find the value of x in the equation 900 = 1000, there being given the logarithm of 3.14 = 0.496930.