# derivative of logarithm with respect to base

The

 $\displaystyle\frac{\partial}{\partial a}\log_{a}x\;=\;-\frac{\ln{x}}{(\ln{a})^% {2}a}$ (1)

for the partial derivative of logarithm expression with respect to the base $a$ may be derived by denoting first

 $\log_{a}x\;=\;y.$

By the definition of logarithm, this equation means the same as

 $a^{y}\;=\;x,$

where we can take the natural logarithms

 $y\ln{a}\;=\;\ln{x}$

solving then

 $y\;=\;\frac{\ln{x}}{\ln{a}}.$

Then, the differentiation is easy:

 $\frac{\partial y}{\partial a}\;=\;\frac{0\ln{a}-\frac{1}{a}\ln{x}}{(\ln{a})^{2% }}\;=\;-\frac{\ln{x}}{(\ln{a})^{2}a}.$
Title derivative of logarithm with respect to base DerivativeOfLogarithmWithRespectToBase 2013-03-22 19:11:28 2013-03-22 19:11:28 pahio (2872) pahio (2872) 4 pahio (2872) Definition msc 26-00 msc 26A09 msc 26A06 Derivative