Answer
a) $0 \lt x \lt 1$
b) $x\gt \ln 5$
Work Step by Step
a) We have $\ln(x)\lt0$.
Since $\ln{(1)}=0$ so the given inequality becomes:
$\ln(x)\lt \ln{(1)}$
Convert the logarithm into exponential form using the fact that $\ln(x)\lt \ln{(y)}$ is equivalent to $x\lt y $.
So $\ln(x)\lt \ln{(1)}$$\to$ $x\lt 1$ $~~~~$$\mathbf (1)$
We know that the domain of the natural logarithm function $\ln(x)$ is:
$x \gt 0$ $~~$$\mathbf(2)$
From $\mathbf(1)$ and $\mathbf(2)$ the given inequality is true if:
$x\lt 1$ $\text{and}$ $x\gt 0$
Find the intersection of the solution:
$x\in \langle 0,1 \rangle$
Write the expression as a compound inequality:
$0 \lt x \lt 1$
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b) $e^{x}\gt5$
Take the natural logarithm of both sides of the inequality:
$\ln{(e^{x})}\gt \ln{(5)}$
Use $\ln{(e^{x})}=x$ to rewrite the left side of the inequality:
$x\gt \ln 5$
