-2ln(x)<1 — step-by-step solution

1) Let's solve in

ℝ

-2 \cdot \ln (x) < 1

We divide both sides by

-2

; since this number is negative, we reverse the inequality sign

\ln (x) > - \frac{1}{2}

Let's apply the exp function on both sides:

e^{\ln (x)} > e^{- \frac{1}{2}}

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (x)}

x

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

e^{- \frac{1}{2}}

\frac{1}{e^{\frac{1}{2}}}

Candidates, pending the domain check:

x > \frac{1}{e^{\frac{1}{2}}}

For the expression

\ln (x)

to exist, you must check:

x > 0

Domain constraints:

x > 0

▶Solution:

x > \frac{1}{e^{\frac{1}{2}}}

2) The numeric value of:

x > \frac{1}{e^{\frac{1}{2}}}

is:

x > \frac{1}{{\approx 2.7183}^{0.5}}

x > \frac{1}{\approx 1.6487}

▷

x > 0.6065 (approximate value)