Confusion about the domain of this logarithmic function

In summary: HallsofIvy!In summary, the function ##f(x)=\ln x^4## has a domain of x ∈ ℝ , x ≠ 0, but if it is changed to ##f(x) = 4 \ln x##, the domain becomes x > 0. This is because the property ##\ln a^b = b\ln a## is only valid for a > 0, and since ##x^4 > 0## if and only if ##x \ne 0##, the same does not hold true for x itself. Therefore, the correct expression for the function is ##f(x)=\ln|x^4|## or ##4\cdot \ln|x|##
  • #1
songoku
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TL;DR Summary
For function ##f(x)=\ln x^4## the domain is x ∈ ℝ , x ≠ 0 but if I change it into ##f(x) = 4 \ln x## then the domain will be x > 0

In my opinion ##\ln x^4## and ##4 \ln x## are two same functions but I am confused why they have different domains
Should I just follow the original question? If given as ##f(x)=\ln x^4## then the domain is x ∈ ℝ , x ≠ 0 and if given as ##f(x) = 4 \ln x## the domain is x > 0? So for the determination of domain I can not change the original question from ##\ln x^4## to ##4 \ln x## or vice versa?

Thanks
 
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  • #2
Same thing goes for ## (x^2)^{1/4}##. this is equal to ##x^{1/2}## only if ##x \geq 0 ##.

songoku said:
So for the determination of domain I can not change the original question from ln⁡x4 to 4ln⁡x or vice versa?
I'd say no.
 
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  • #3
songoku said:
Summary:: For function ##f(x)=\ln x^4## the domain is x ∈ ℝ , x ≠ 0 but if I change it into ##f(x) = 4 \ln x## then the domain will be x > 0

In my opinion ##\ln x^4## and ##4 \ln x## are two same functions but I am confused why they have different domains

Should I just follow the original question? If given as ##f(x)=\ln x^4## then the domain is x ∈ ℝ , x ≠ 0 and if given as ##f(x) = 4 \ln x## the domain is x > 0? So for the determination of domain I can not change the original question from ##\ln x^4## to ##4 \ln x## or vice versa?

Thanks
The solution is: you cheated!

If we write ##g(x)=x^4## then ##f=\ln\circ g## which is only defined if we use absolute values: ##f=\ln\circ \operatorname{abs} \circ g##. So the correct expression is ##f(x)=\ln|x^4|## which equals ##4\cdot \ln|x|##. The fact that you could omit the absolute value is due to your unmentioned knowledge that ##x^4\geq 0## for all ##x##. Hence you used an additional information which was hidden, whereas the camouflage vanished in ##\ln x##.
 
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  • #4
songoku said:
For function ##f(x)=\ln x^4## the domain is x ∈ ℝ , x ≠ 0 but if I change it into ##f(x) = 4 \ln x## then the domain will be x > 0

In my opinion ##\ln x^4## and ##4 \ln x## are two same functions but I am confused why they have different domains
The property of logarithms that you used, ##\ln a^b = b\ln a## is valid only for a > 0. ##x^4 > 0## if and only if ##x \ne 0##, but the same is not true for x itself.
 
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  • #5
Thank you very much for the help dRic2, fresh_42, Mark44
 
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1. What is a logarithmic function?

A logarithmic function is a type of mathematical function that relates the input value (x) to the output value (y) by using the logarithm of the input value. It is written in the form y = logb(x), where b is the base of the logarithm.

2. What is the domain of a logarithmic function?

The domain of a logarithmic function is the set of all possible input values (x) for which the function is defined. For a logarithmic function, the domain is all positive real numbers, as the logarithm of a negative number is undefined.

3. How do I determine the domain of a logarithmic function?

To determine the domain of a logarithmic function, you need to consider the base of the logarithm. If the base is greater than 1, the domain is all positive real numbers. If the base is between 0 and 1, the domain is all positive real numbers except 0. If the base is 1, the domain is undefined.

4. Can the domain of a logarithmic function be negative?

No, the domain of a logarithmic function cannot be negative. As mentioned earlier, the logarithm of a negative number is undefined, so the domain must be limited to positive real numbers.

5. What happens if the input value of a logarithmic function is 0?

If the input value of a logarithmic function is 0, the function is undefined. This is because the logarithm of 0 is undefined in all bases. Therefore, 0 is not included in the domain of a logarithmic function.

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