ln(x^2)>2 — step-by-step solution

1) Let's solve in

ℝ

\ln (x^{2}) > 2

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

\ln (x^{2}) > 2

is equivalent to

x^{2} > e^{2}

An even power only depends on the absolute value:

x^{2} > e^{2}

is equivalent to

| x | > e

| A | > B

means that:

A > B

A < - B

Candidates, pending the domain check:

x \in (-∞, - e) \cup (e, +∞)

2) Let's find the domain of

\ln (x^{2}) > 2

For the expression

\ln (x^{2})

to exist, you must check:

x^{2} > 0

In ℝ, an even exponent always yields a value ≥ 0 and it is 0 only if the base is 0

Domain constraints:

x \neq 0

▶Solution:

x \in (-∞, - e) \cup (e, +∞)

3) The numeric value of:

x \in (-∞, - e) \cup (e, +∞)

equals:

▷

x \in (-∞, \approx -2.7183) \cup (\approx 2.7183, +∞)