B Confusion about the domain of this logarithmic function

AI Thread Summary
The discussion centers on the domains of the logarithmic functions f(x) = ln(x^4) and f(x) = 4 ln(x). For f(x) = ln(x^4), the domain is x ∈ ℝ, x ≠ 0, while for f(x) = 4 ln(x), the domain is restricted to x > 0. The confusion arises from the assumption that these two forms represent the same function, but they do not due to the properties of logarithms. The logarithmic property ln(a^b) = b ln(a) is valid only when a > 0, which explains the differing domains. Ultimately, the original question should be followed to accurately determine the domain of each function.
songoku
Messages
2,470
Reaction score
386
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
 
Mathematics news on Phys.org
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.
 
  • Like
Likes songoku
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##.
 
  • Like
Likes Adesh, songoku and dRic2
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.
 
  • Like
Likes Stephen Tashi, Delta2 and songoku
Thank you very much for the help dRic2, fresh_42, Mark44
 
  • Like
Likes dRic2
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top