Integral of Logarithmic Function to Any Base : Log_a (x)

 

The steps to find the integral of a logarithmic function to any base are presented.

Use of the Change of Base Formula

Let $y = \log_a x$
Use the
change of base formula to rewrite $y = \log_a x$ using the natural logarithm $\ln$ as
$y = \log_a x = \dfrac{\ln x}{\ln a}$

We now evaluate the integral
$\displaystyle \int \log_a x \; dx = \int \left(\dfrac{ \ln x }{\ln a}\right)\; dx$

$\ln a$ is a constant and therefore
$\displaystyle \int \log_a x \; dx = \dfrac{ 1}{\ln a} \int \ln x \; dx \qquad (I)$

The
integral of ln x is given by
$\displaystyle \int \ln x \; dx = x \ln x - x + c$

Substitute in $(I)$ to obtain

$\displaystyle \int \log_a x \; dx = \dfrac{ 1}{\ln a} (x \ln x - x) + c$