How do you find the domain of $C(x)=ln((x+3)^4 )$ ?

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Answer 1 · 2016-09-14T22:07:48

The domain of $C(x)$ is $(-oo, -3) uu (-3, oo)$

Assuming we're dealing with the Real natural logarithm, the domain is $RR "\" { -3 }$

When $x = -3$ :

$(x+3)^4 = ((-3)+3)^4 = 0^4 = 0$

and $ln(0)$ is (always) undefined.

So $-3$ is not in the domain.

When $x != -3$

$(x+3)^4 > 0$

so $ln((x+3)^4)$ is well defined.

So the domain is the whole of the Real numbers except $-3$ .

In interval notation $(-oo, -3) uu (-3, oo)$

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Footnote

The interesting thing about this question is the immediate temptation to turn:

$C(x) = ln((x+3)^4)$

into:

$C(x) = 4 ln(x+3)$

While this would be true for any $x > -3$ , it is not true for $x < -3$ .

In fact, if you extend the definition of $ln$ to Complex values, allowing the logarithm of negative numbers, then if $t < 0$ we have:

$ln t = ln abs(t) + pi i" "$ (principal value)

and hence if $x < -3$ :

$4 ln (x+3) = 4 (ln abs(x+3) + pi i) = 4 ln abs(x+3) + 4 pi i != 4 ln abs(x+3) = ln ((x+3)^4)$

How do you find the domain of C(x)=ln((x+3)^4 )C(x)=ln((x+3)^4 )?