High School Confusion about the domain of this logarithmic function

[songoku]

TL;DR

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

 

[dRic2]

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.

 

[fresh_42]

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 $\ln x$.

 

[Mark44]

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.

 

[songoku]

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