372 Professor Hamitton on Conjugate Functions, 
And we may give the name of exponents or logarithms to the determining numbers, 
ordinal or cardinal, 
SEO S402, Ole Otel et Sone, (266.) 
which answer the question “ which in order is the Power?” or this other question 
‘« Have any (effective) acts of multiplication, equivalent or reciprocal to the original 
act of multiplying by the given ratio a, been combined to produce the act of multi- 
plying by the Power ; and if any, then How many, and Jn which direction, that is, 
whether are they similar or opposite in effect, (as enlarging or diminishing the step on 
which they are performed,) to that original act?” Thus 2 is the logarithm of the 
square or second power aa, when compared with the base a; 3 is the logarithm of 
the cube aaa, 1 is the logarithm of the base a itself, © 1 is the logarithm of the 
reciprocal u a, and O is the logarithm of the ratio 1 considered as the zero-power 
of a. 
With these conceptions and notations of powers and logarithms, we can easily 
prove the relation 
a’ x a“=za rte, (267.) 
for any integer logarithms » and v, whether positive, or contra-positive, or null ; and 
this other connected relation 
b’ =a’ X+ if b=ae; 268.) 
which may be thus expressed in words: ‘‘ Any two powers of any common base may 
be multiplied together by adding their logarithms,’ and ‘‘ Any proposed power may 
be powered by any proposed whole number, by multiplying its logarithm by that 
number,” if the sum of the two proposed logarithms in the first case, or the multiple 
of the proposed logarithm in the second case, be employed as a new logarithm, to 
form a new power of the original base or ratio; the logarithms here considered being: 
all whole numbers. 
30. The act of passing from a base to a power, is connected with an inverse or 
reciprocal act of returning from the power to the base ; and the conceptions of both 
these acts are included in the more comprehensive conception of the act of passing 
from any one to any other of the ratios of the series (261.) or (262.). This act of 
passing from any one power a“ to any other power a” of a common base a, may be 
still called in general an act of powering ; and more particularly, (keeping up the 
analogy to the language already employed in the theory of multiple steps,) it may be 
called the act of powering by the fractional number ae By the same analogy of 
I 
