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Natural logarithm definitions

Submitted by admin on Wed, 01/05/2011 - 11:08

Natural logarithm

 

e is the natural logarithm,

e = 2.718281828.

e^{\ln(x)} = x \qquad \mbox{if }x > 0\,\!

 

\ln(e^x) = x.\,\!

 

\ln(xy) = \ln(x) + \ln(y) \!\,

 

\ln(1) = 0\,

\ln(-1) = i \pi \quad \,

 

\ln(a)=\int_1^a \frac{1}{x}\,dx.

 

\ln(ab)=\ln(a)+\ln(b) \,\!

 

\frac{d}{dx}\log_b(x) = \frac{\log_b(e)}{x} =\frac{1}{x\ln(b)}

\ln (ab)<br /> = \int_1^{ab} \frac{1}{x} \; dx<br /> = \int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx<br /> =\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt<br /> = \ln (a) + \ln (b)

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