ln(x^4)<4 — step-by-step solution

1) Let's solve in

ℝ

\ln (x^{4}) < 4

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

\ln (x^{4}) < 4

is equivalent to

x^{4} < e^{4}

x^{4} < e^{4}

An even power only depends on the absolute value:

x^{4} < e^{4}

is equivalent to

| x | < e

| x | < e

| A | < B

is equivalent to

- B < A < B

- e < x < e

Candidates, pending the domain check:

x \in (- e, e)

2) Let's evaluate the domain of

\ln (x^{4}) < 4

For the expression

\ln (x^{4})

to exist, you must check:

x^{4} > 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 (- e, 0) \cup (0, e)

3) The numeric value of:

x \in (- e, 0) \cup (0, e)

is:

▷

x \in (\approx -2.7183, 0) \cup (0, \approx 2.7183)