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bergausstein
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Show that ∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C
can you show me the complete solution to this prob? thanks!
can you show me the complete solution to this prob? thanks!
From the laws of logarithms:\(\displaystyle \log_b x=\frac{\log_c x}{\log_c b}\)bergausstein said:Show that ∫ logb(x) dx = x ∙ ( logb(x) - 1 / ln(b) ) + C
can you show me the complete solution to this prob? thanks!
The notation "∫ logb(x) dx" represents the indefinite integral of the logarithmic function with base b. It is the inverse operation of finding the derivative of a function.
C is the constant of integration and its exact value depends on the initial conditions of the problem. It is added to the result of the indefinite integral to account for all possible solutions.
One way to show this is by using the power rule for integration and the logarithmic properties. By rewriting logb(x) as ln(x) / ln(b), we can apply the power rule and solve for the integral. The constant of integration can also be added to the final result.
The ln(b) term is the natural logarithm of the base b. It appears in the result of the integration because the logarithmic function is defined using the natural logarithm. It also plays a crucial role in the properties of logarithms and their derivatives.
This equation can be applied to any base b as long as the logarithmic function is defined for that base. However, the properties of logarithms and their derivatives may vary for different bases and may require different methods of integration.