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:

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

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