For powers or roots other than those of the simple forms already
discussed, it is necessary to employ the usual logarithmic process. Thus
to find _a^n_ = _x_, we multiply the logarithm of _a_ by _n_, and find
the number _x_ corresponding to the logarithm so obtained. Similarly, to
find _ⁿ√̅a_ = _x_ we divide the logarithm of _a_ by _n_, and find the
number _x_ corresponding to the resulting logarithm.
_The Scale of Logarithms._—Upon the back of the slide of the Gravêt and
similar slide rules there will be found three scales. One of
these—usually the centre one—is divided equally throughout its entire
length, and figured from right to left. It is sometimes marked L,
indicating that it is a scale giving logarithms. The whole scale is
divided primarily into ten equal parts, and each of these subdivided
into 50 equal parts. In the recess or notch in the right-hand end of the
rule is a reference mark, to which any of the divisions of this
evenly-divided scale can be set.
As this decimally-divided scale is equal in length to the logarithmic
scale D, and is figured in the reverse direction, it results that when
the slide is drawn to the right so that the L.H. index of C coincides
with any number on D, the reading on the equally-divided scale will give
the decimal part of the logarithm of the number taken on D. Thus if the
L.H. index of C is placed to agree with 2 on D, the reading of the back
scale, taken at the reference mark, will be found to be 0·301, the
logarithm of 2. It must be distinctly borne in mind that the number so
obtained is the _decimal part_ or _mantissa_ of the logarithm of the
number, and that to this the characteristic must be prefixed in
accordance with the usual rule—viz., _The integral part, or
characteristic of a logarithm is equal to the number of digits in the
number, minus 1. If the number is wholly decimal, the characteristic is
equal to the number of cyphers following the decimal point, plus 1._ In
the latter case the characteristic is negative, and is so indicated by
having the minus sign written _over_ it.
To obtain any given power or root of a number, the operation is as
follows:—Set the L.H. index of C to the given number on D, and turning
the rule over, read opposite the mark in the notch at the right-hand end
of the rule, the decimal part of the logarithm of the number. Add the
characteristic according to the above rule, and multiply by the exponent
of the power, or divide by the exponent of the root. Place the _decimal
part_ of the resultant reading, taken on the scale of equal parts,
opposite the mark in the aperture of the rule, and read the answer on D
under the L.H. index of C, pointing off the number of digits in the
answer in accordance with the number of the characteristic of the
resultant.
EX.—Evaluate 36^{1·414}.
Set 1 on C to 36 on D and read the decimal part of log. 36 on the
scale of logarithms on the back of the slide. This value is found to
be 0·556. As there are two digits in the number, the characteristic
will be 1; hence log. 36 = 1·556. Multiply by 1·414, using the C and D
scales, and obtain 2·2 as the log. of the result. Set the decimal
part, 0·2, on the log. scale to the mark in the notch at the end of
the rule and read 1585 on D under 1 on C. Since the log. of the result
has a characteristic 2, there will be 3 digits in the result, which is
therefore read as 158·5.
This example will suffice to show the method of obtaining the nth power
or the _n_th root of _any_ number.