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The index z is called the Logarithm of the num- 
ber X. 
2. Hence, to find the logarithm z of any num- 
ber X, is only to find what power of the ‘radijca! 
number r, in that fcale, is equal' to the number x ; 
or to find the index z of the power, in the equation 
X —r^. 
3. The properties of logarithms are the fame with, 
the indices of powers ; that is, thp fpm or difference 
of the logarithms of two numbers, is the logarithm 
of the produdl or quotient of thofe numbers. 
And therefore, ?i times the Jogarithm of any num- 
ber, is the logarithm of the nih. power of that number.. 
4. The relation of any number Xy and its loga- 
rithm z being given; To find the relation of their 
leaft fynchronal variation x and z 
Put 1 radical number of any fcale, and 
n 
9=T+-n 
Let j y — ~- 
Then / x—xz fliews the relation required,. 
For x = r'^=i-{-r^"'» 
Now, let X and z flow lb that x becomes x~^x, at 
the fame time as z fhall become 
Then x x= i i x i 
=xy. i-\-zq-\-lzf -\-^zq^-\-\zq*y^c. 
Therefore x=zxz'aq-\-lq^-\--^q' -{-\q*y bcc. 
7=zxza=:x zx~* Confequently /'x=zxz. 
