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

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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

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