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)#

#color(white)()#
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)#