How to find integral of logarithmic functions

How to find integral of logarithmic functions:

$$\boxed{\int{log_a(x) \ dx} = x * log_a(x) - \dfrac{x}{\ln{(a)}} + c\ }$$

$$\begin{array}{lcrclcrccccl}
(u & * & v)' & = & u' &*& v & + & u &*& v' &\\
(x & * & log_a{(x)})' & = & 1 & *& log_a{(x)} & + & x & * & \dfrac{1}{x*\ln{(a)}} & \quad | \quad \textcolor[rgb]{1,0,0}
{
( log_a{(x)} )' = \dfrac{1}{x*\ln{(a)}}
}\\ \\
(x & * & log_a{(x)})' & = &&& log_a{(x)} & + &&& \dfrac{1}{\ln{(a)}} & \quad | \quad \int{}\ dx \\ \\
x & * & log_a{(x)} & = &&& \int{ log_a{(x)} \ dx} & + &&& \dfrac{1}{\ln{(a)}} \int{\ dx} & \\ \\
x & * & log_a{(x)} & = &&& \int{ log_a{(x)} \ dx} & + &&& \dfrac{x}{\ln{(a)}} & \\ \\
\int{ log_a{(x)} \ dx} & & & = &&& x * log_a{(x)} & - &&& \dfrac{x}{\ln{(a)}} &
\end{array}$$

Example:

$$basic \ a = e: \\
\int{ log_e{(x)} \ dx} =x * log_e{(x)} - \dfrac{x}{\ln{(e)}} \\
\int{ \ln{(x)} \ dx} =x * ln{(x)} - \dfrac{x} {1} \\
\int{ \ln{(x)} \ dx} =x * ln{(x)} - x\\$$

$$basic \ a = 10: \\
\int{ log_{10}{(x)} \ dx} =x * log_{10}{(x)} - \dfrac{x}{\ln{(10)}} \\
\int{ \log{(x)} \ dx} =x * log{(x)} - \dfrac{x}{\ln{(10)}}$$