ln(x^3)<3 — step-by-step solution

1) Let's solve in

ℝ

\ln (x^{3}) < 3

On the logarithm's domain, ln is strictly increasing:

\ln (x^{3}) < 3

is equivalent to

x^{3} < e^{3}

x^{3} < e^{3}

The function

x \mapsto x^{3}

being strictly increasing on

ℝ

x^{3} < e^{3}

is equivalent to

x < \sqrt[3]{e^{3}} = e

Candidates, pending the domain check:

x < e

2) Let's evaluate the domain of

\ln (x^{3}) < 3

For the expression

\ln (x^{3})

to exist, you must check:

x^{3} > 0

x^{3} > 0

is equivalent to:

x > 0

Domain constraints:

x > 0

▶Solution:

x \in (0, e)

3) The numeric value of:

x \in (0, e)

equals:

▷

x \in (0, \approx 2.7183)