Integrating a logarithmic function

In summary, the conversation is about the lack of a formula for finding the integral of log base a of x and the solution is to use the formula \log_a{x} = \frac{\ln{x}}{\ln{a}} or to use change of base and then integrate using substitution. The conversation ends with a compliment to the expert and a prediction of their success at CalTech.
  • #1
Math Is Hard
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I noticed than in the chapters I am studying now that while they give us a formula for taking the derivative of log base a of x, I can't find a correspoding formula for finding the integral of log base a of x.
We have a table of integrals in the back of the book, but I only see integrals pertaining to forms of ln x.
So.. am I missing something? Does it not exist? Is it something so ugly I don't even want to know about it??

Thanks!
 
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  • #2
[tex]\log_a{x} = \frac{\ln{x}}{\ln{a}}[/tex]

cookiemonster
 
  • #3
ok, so I use change of base to convert. Then to integrate, do I use substitution, or is it simpler than that? thanks.
 
  • #4
ln(a) is a constant. It comes right out front.

cookiemonster
 
  • #5
Oh crud! I walked away from the computer after my last post and then all of a sudden it hit me like a ton of bricks! :eek:
thanks, CM. You are going to be a big hit at CalTech. :biggrin:
 

1. What is a logarithmic function?

A logarithmic function is a mathematical function that represents the inverse of an exponential function. It is written in the form f(x) = logb(x), where b is the base of the logarithm. It is commonly used to model relationships between quantities that change exponentially.

2. How do you integrate a logarithmic function?

To integrate a logarithmic function, you can use the power rule for logarithms, which states that ∫logb(x)dx = xlogb(x) - x + C, where C is the constant of integration. This rule can also be extended to include logarithmic functions with different bases, such as ln(x) or log2(x).

3. What is the purpose of integrating a logarithmic function?

Integrating a logarithmic function allows us to find the area under the curve of the function. This can be useful in various applications, such as calculating the growth rate of a population or determining the rate of decay of a substance.

4. Are there any special techniques for integrating complicated logarithmic functions?

Yes, there are several techniques that can be used to integrate more complex logarithmic functions. These include using substitution, integration by parts, and partial fraction decomposition. It is important to carefully analyze the function and choose the most appropriate technique for integration.

5. Can logarithmic functions be integrated over a range of values?

Yes, logarithmic functions can be integrated over a range of values by using definite integration. This involves evaluating the integral at the upper and lower limits of the range and subtracting the two values. This will give the exact area under the curve of the logarithmic function within the given range.

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