The steps to find the integral of a logarithmic function to any base are presented below.
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 (\( \ln \)):
\[ y = \log_a x = \dfrac{\ln x}{\ln a} \]We now evaluate the integral:
\[ \int \log_a x \, dx = \int \left(\dfrac{\ln x}{\ln a}\right) dx \]Since \( \ln a \) is a constant, we can factor it out:
\[ \int \log_a x \, dx = \dfrac{1}{\ln a} \int \ln x \, dx \qquad (I) \]The integral of \( \ln x \) is given by:
\[ \int \ln x \, dx = x \ln x - x + c \]Substitute this back into equation (I) to obtain:
Integral Formula for \( \log_a x \):
\[ \int \log_a x \, dx = \dfrac{1}{\ln a} (x \ln x - x) + c \]