Question about logarithmic differentiation

In summary, the conversation discusses logarithmic differentiation and the issue of the absolute value involved. The example provided is ##y = 2^x \sin x##, with a domain of all real numbers. Taking the natural logarithm of both sides results in gaps in the graph, as the domain of ln is positive reals and the function is not defined for intervals where sin(x) is negative. The solution involves making the question more specific and considering only real-valued functions.
  • #1
Mr Davis 97
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I have a question about logarithmic differentiation, especially concerning the absolute value involved. For example, if we have the function ##y = 2^x \sin x##, the domain is all real numbers. So what happens when we take ##\ln## of both sides of the equation? The antilogarithm must be greater than ##0##, but the domain of the function is all real numbers. How do we account for this? I know the answer has something to do with absolute value, but I am not sure how...
 
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  • #2
Ln of negative numbers gets you into complex analysis. I suggest you make your question more specific.
 
  • #3
Assuming that you're concerned only with real-valued functions, since the domain of ln is positive reals, ##\ln(2^x\sin(x))## is defined only for intervals in which sin(x) > 0, open intervals of the form ##(2k\pi, (2k + 1)\pi)##, with k in the integers. For the other intervals, ##\ln(2^x\sin(x))## is not defined, resulting in gaps in the graph.
 

Related to Question about logarithmic differentiation

1. What is logarithmic differentiation?

Logarithmic differentiation is a method used to find the derivative of a function that is in the form of a logarithm. It involves taking the natural logarithm of both sides of the function, using logarithmic properties to simplify it, and then using the chain rule to find the derivative.

2. When is logarithmic differentiation useful?

Logarithmic differentiation is useful when you have a complicated function that is difficult to differentiate using traditional methods, such as the product, quotient, or chain rule. It can also be useful when you have a function that involves both exponential and logarithmic terms.

3. How do you perform logarithmic differentiation?

To perform logarithmic differentiation, you first take the natural logarithm of both sides of the function. Then, you use logarithmic properties such as the power rule and product rule to simplify the function. Finally, you use the chain rule to find the derivative of the function.

4. What is the difference between logarithmic differentiation and regular differentiation?

The main difference between logarithmic differentiation and regular differentiation is the method used to find the derivative. Regular differentiation uses traditional methods such as the power rule, product rule, and chain rule, while logarithmic differentiation involves taking the natural logarithm of the function and using logarithmic properties to simplify it.

5. Are there any limitations to using logarithmic differentiation?

While logarithmic differentiation can be useful in certain situations, it also has its limitations. It can only be used for functions that are in the form of a logarithm, and it may not always be the most efficient method for finding the derivative. It is important to consider other methods of differentiation before resorting to logarithmic differentiation.

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