ln(x)/ln(2)>1 — step-by-step solution

1) Let's solve in

ℝ

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

2) Let's study the sign of:

\ln (2)

\approx 0.6931

\ln (x) > \ln (2)

ⓘ We subtract

\ln (2)

from both sides

\ln (x) - \ln (2) > 0

ⓘ For strictly positive arguments, a sum of logarithms is the logarithm of the product, a difference that of the quotient, ln(a) + ln(b) = ln(a*b):

\ln (x) - \ln (2)

\ln (\frac{x}{2})

\ln (\frac{x}{2}) > 0

You can apply the exp function on both sides:

e^{\ln (\frac{x}{2})} > e^{0}

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (\frac{x}{2})}

\frac{x}{2}

■Any nonzero number raised to the power 0 is 1:

e^{0}

1

\frac{x}{2} > 1

Candidates, pending the domain check:

x > 2

For the expression

\ln (x)

to be defined, we must have:

x > 0

Domain constraints:

x > 0

▶Solution:

x > 2