dfrac {x}{\ln \!\alpha }}\end{cases}}} - Wikimedia

{\displaystyle {\begin{cases}\int \ln \!x\ {\mbox{d}}x&=x\ln \!x-\int x\!{\dfrac {1}{x}}\ {\mbox{d}}x\\&=x\ln \!x-\int \!\!{\mbox{d}}x\\&=x\ln \!x-x\\\int ... {\displaystyle{\begin{cases}\int\ln\!x\{\mbox{d}}x&=x\ln\!x-\intx\!{\dfrac{1}{x}}\{\mbox{d}}x\\&=x\ln\!x-\int\!\!{\mbox{d}}x\\&=x\ln\!x-x\\\int\ln\!-x\{\mbox{d}}x&=-\intx\ln\!-x\{\mbox{d}}\!-\!x\\&=-\left(\int-x\ln\!-x+x\right)\\&=x\ln\!-x-x\end{cases}}\therefore{\begin{cases}\int\ln\!|x|&=x\ln\!|x|-x\\\int\log_{\alpha}\!|x|{\mbox{d}}x&=\int{\dfrac{\ln\!|x|}{\ln\!\alpha}}{\mbox{d}}x\\&={\dfrac{x\ln\!|x|-x}{\ln\!\alpha}}\\&=x\log_{\alpha}\!|x|-{\dfrac{x}{\ln\!\alpha}}\end{cases}}}