What is the domain of $arcsin (ln (x))$ $arcsin(Ln(x))$ ?

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Answer 1 · 2016-08-04T10:37:22

Let $y = arcsin (ln x)$ $y=arcsin(lnx)$ hence $sin y = ln x$ $siny=lnx$

But the domain of $ln x$ $lnx$ is $x > 0$ $x>0$

The domain of siny is $- 1 \leq sin y \leq 1$ $-1<=siny<=1$ hence

$- 1 \leq ln x \leq 1$ $-1<=lnx<=1$ hence $e^{- 1} \leq x \leq e^{1}$ $e^-1<=x<=e^1$

Finally the domain of $y = arcsin (ln x)$ $y=arcsin(lnx)$ is $D_{y} = [\frac{1}{e}, e]$ $D_y=[1/e,e]$

What is the domain of arcsin(ln(x))arcsin(Ln(x))?