Effortlessly Solve Integrals: Learn the Derivative of Logarithmic Functions

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In summary, the derivative of \log_{e} 4x is \frac{1}{x} and the correct answer to the integral \int \frac{1}{x\log_{e} 4x} dx is \log_{e}(\log_{e}4x)+c. The substitution u = \ln 4x was used to solve the integral.
  • #1
meee
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[tex]\int \frac{1}{x\log_{e} 4x} dx

[/tex]

wow i finally got latex to work

ok... so i did it on a text and got [tex]\log_{e}(\log_{e}4x)+c[/tex]

now i did it and got [tex]\frac{1}{4}\log_{e}(\log_{e}4x)+c[/tex]

which is right?

what is the derivative of [tex]\log_{e}4x[/tex] ?
 
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  • #2
do you mean [tex] \int \frac{1}{x\ln 4x} dx [/tex]? Use the substitution [tex] u = \ln 4x [/tex].
 
  • #3
your first answer is correct. can you see why?

[tex] \log_{e} 4x = \ln 4x [/tex]. Can you take the derivative of that now?
 
  • #4
courtrigrad said:
your first answer is correct. can you see why?

[tex] \log_{e} 4x = \ln 4x [/tex]. Can you take the derivative of that now?
ohhhh... is it [tex]\frac{4}{4x} = \frac{1}{x} [/tex] ?
 
  • #5
yes it is.
 
  • #6
i see.. Thanks
 

FAQ: Effortlessly Solve Integrals: Learn the Derivative of Logarithmic Functions

What are logarithmic functions?

Logarithmic functions are mathematical functions that involve the logarithm of a variable. The logarithm of a number is the power to which a given base must be raised to produce that number. In other words, it is the inverse function of exponentiation.

How do logarithmic functions relate to integrals?

Logarithmic functions are important in calculus because they have derivative of the form 1/x. This means that when we integrate a logarithmic function, we end up with an expression that involves the natural logarithm. Understanding the derivative of logarithmic functions is essential for solving integrals involving these types of functions.

What is the derivative of a logarithmic function?

The derivative of a logarithmic function f(x) is given by the formula f'(x) = 1/x. This means that the slope of the tangent line to the graph of a logarithmic function at any point is equal to 1 divided by the x-coordinate of that point. In other words, the derivative of a logarithmic function is the reciprocal of the input value.

How can I use the derivative of logarithmic functions to solve integrals?

When solving integrals involving logarithmic functions, we can use the derivative of logarithmic functions to simplify the expression and make it easier to integrate. By understanding the relationship between logarithmic functions and their derivatives, we can use known integration techniques such as substitution or integration by parts to solve these types of integrals.

Are there any shortcuts for solving integrals of logarithmic functions?

Yes, there are several techniques that can help make solving integrals of logarithmic functions easier. These include using the properties of logarithms, recognizing common patterns, and using known formulas for specific types of integrals. However, it is important to have a solid understanding of the derivative of logarithmic functions in order to effectively use these shortcuts.

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