ABSTRACT
Logarithmic differentiation is achieved with knowledge of (i) the laws of logarithms, (ii) the differential coefficients of logarithmic functions, and (iii) the differentiation of implicit functions.
The laws of logarithms are:
log ( A × B ) = log A + log B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3460.tif"/>
log ( A B ) = log A − log B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3461.tif"/>
log A n = n log A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3462.tif"/>
The differential coefficient of the logarithmic function ln x is given by: d d x ( l n x ) = 1 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3463.tif"/>
More generally, it may be shown that: () d d x [ l n f ( x ) ] = f ′ ( x ) f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3464.tif"/>
Differentiation of implicit functions is obtained using: () d dx [ f ( y ) ] = d dy [f(y)] × dy dx https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3465.tif"/>
