Confusion about the domain of this logarithmic function

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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

Main Question or Discussion Point

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
 

Answers and Replies

  • #2
dRic2
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Same thing goes for ## (x^2)^{1/4}##. this is equal to ##x^{1/2}## only if ##x \geq 0 ##.

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
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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
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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
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Thank you very much for the help dRic2, fresh_42, Mark44
 
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