Primitive of General Logarithm Function

Theorem

\(\ds \int \log_a x \rd x\) \(=\) \(\ds \dfrac 1 {\ln a} \paren {x \ln x - x} + C\)
\(\ds \) \(=\) \(\ds x \log_a x - \dfrac x {\ln a}\)


Proof

\(\ds \int \log_a x \rd x\) \(=\) \(\ds \int \dfrac {\ln x} {\ln a} \rd x\) Change of Base of Logarithm
\(\ds \) \(=\) \(\ds \dfrac 1 {\ln a} \paren {x \ln x - x} + C\) Primitive of $\ln x$
\(\ds \) \(=\) \(\ds \dfrac {x \ln x} {\ln a} \dfrac x {\ln a} + C\) rearranging
\(\ds \) \(=\) \(\ds x \log_a x - \dfrac x {\ln a} + C\) Change of Base of Logarithm

$\blacksquare$


Sources

  • 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $11$.
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