ln(x^2)<=0 — step-by-step solution

Let's solve in

ℝ

\ln (x^{2}) \leq 0

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

\ln (x^{2}) \leq 0

is equivalent to

x^{2} \leq 1

An even power only depends on the absolute value:

x^{2} \leq 1

is equivalent to

| x | \leq 1

| A | \leq B

is equivalent to

- B \leq A \leq B

-1 \leq x \leq 1

Candidates, pending the domain check:

x \in [-1, 1]

Let's find the domain of

\ln (x^{2}) \leq 0

For the expression

\ln (x^{2})

to be defined, we must have:

x^{2} > 0

Any even power is always non-negative in ℝ and equals 0 only when the base itself is 0

Domain constraints:

x \neq 0

▶Solution:

x \in [-1, 0) \cup (0, 1]