I'm confused trying to figure out the domain of the logarithmic function below: f ( x ) = ln ⁡ ( e x + 3 )Because the argument of f , e x + 3 ,, is a nonnegative number, e x + 3 &gt; 0 and e x &gt; − 3. Taking natural logarithms on both sides, we get x &gt; ln ⁡ ( − 3 ). However, the domain of a logarithmic function must be nonnegative real numbers, so ln ⁡ ( − 3 ) doesn't make sense. How then to determine the scope of the original function?

Question

I&#039;m confused trying to figure out the domain of the logarithmic function below:  f  (  x  )  =  ln  ⁡  (      e    x    +  3  )Because the argument of   f  ,        e    x    +  3  ,, is a nonnegative number,       e    x    +  3  &amp;gt;  0 and       e    x    &amp;gt;  −  3. Taking natural logarithms on both sides, we get   x  &amp;gt;  ln  ⁡  (  −  3  ). However, the domain of a logarithmic function must be nonnegative real numbers, so   ln  ⁡  (  −  3  ) doesn&#039;t make sense. How then to determine the scope of the original function?

neglegir86 · Accepted Answer

The domain in interval notation is   (  −  ∞  ,  ∞  ), or   x  ∈      R  You are trying to find out when       e    x    +  3  ≤  0, so you can state restrictions in the domain.      e    x    ≤  −  3    ⟹    x  ≤  ln  ⁡  (  −  3  )Since   ln  ⁡  (  −  3  ) is not defined for real numbers, there are no restrictions.

erkentrs · Accepted Answer

e    x    +  3  &amp;gt;  0  for all   x  ∈      R  , so   log  ⁡  (      e    x    +  3  ) is defined for all   x  ∈      R

Find out the domain of the logarithmic function below: f(x)=ln(e^x+3)

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