Numbers 3 0·3 0·03 0·003 0·0003 etc. Logarithms 0·477 ̅1·477 ̅2·477 ̅3·477 ̅4·477 „ Here again we see that with the same significant figures in the numbers, the mantissa of the logarithm has always the same (_positive_) value, but the characteristic is _one more_ than the _number of 0’s immediately following the decimal point_, and is _negative_, as indicated by the minus sign written over it. Only the decimal parts of the logarithms of numbers between 1 and 10 are given in the usual tables, for, as shown above, the logarithms of all tenfold multiples or submultiples of a number can be obtained at once by modifying the characteristic in accordance with the rules given. An examination of the two rows of figures giving the logarithms of numbers from 1 to 10 will reveal some striking peculiarities, and at the same time serve to illustrate the principle of logarithmic calculation. First, it will be noticed that the addition of any two of the logarithms gives the logarithm of the _product_ of these two numbers. Thus, the addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference of the logarithms of two numbers gives the logarithm of the _quotient_ resulting from the division of these two numbers. Thus, log. 8 − log. 2 = 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2. One other important point is to be noted. If the logarithm of any number is _multiplied_ by 2, 3, or any other quantity, whole or fractional, the result is the logarithm of the original number, raised to the 2nd, 3rd, or other power respectively. Thus, multiplying the log. of 3 by 2, we obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is, of 3 raised to the 2nd power, or 3 _squared_. Again, log. 2 multiplied by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or 2 _cubed_. Conversely, dividing the logarithm of any original number by any number _n_, we obtain the logarithm of the _n_th root of the original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore equal to log. 2 or to the log. of the _cube root_ of 8. Only simple logs. have been taken in these examples, but the student will understand that the same reasoning applies, whatever the number. Thus for 20^3 we prefix the characteristic (1 in this case) to log. 2, giving 1·301. Multiplying by 3, we have 3·903 as the resulting logarithm, and as its characteristic is 3, we know that it corresponds to the number 8000. Hence 20^3 = 8000.