D’fhonn an bonn a athrú ó b go c, is féidir linn an riail maidir le hathrú logartamach a úsáid. Tá bun b logarithm x cothrom le bun c logarithm x roinnte ar bhonn c logarithm b:
log b ( x ) = log c ( x ) / log c ( b )
log 2 (100) = log 10 (100) / log 10 (2) = 2 / 0.30103 = 6.64386
log 3 (50) = log 8 (50) / log 8 (3) = 1.8812853 / 0.5283208 = 3.5608766
Má dhéantar b a ardú le cumhacht bhonn b logarithm x tugtar x:
(1) x = b log b ( x )
Má dhéantar c a ardú le cumhacht bhonn c logarithm b, tugtar b:
(2) b = c log c ( b )
Nuair a thógann muid (1) agus c log c ( b ) (2) in ionad b , faighimid:
(3) x = b log b ( x ) = ( c log c ( b ) ) log b ( x ) = c log c ( b ) × log b ( x )
Trí logáil c () a chur i bhfeidhm ar dhá thaobh (3):
log c ( x ) = log c ( c log c ( b ) × log b ( x ) )
Trí riail na cumhachta logarithm a chur i bhfeidhm :
log c ( x ) = [log c ( b ) × log b ( x )] × log c ( c )
Ó logáil c ( c ) = 1
log c ( x ) = log c ( b ) × log b ( x )
Nó
log b ( x ) = log c ( x ) / log c ( b )
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