ln(x-1)=ln(2-x) — step-by-step solution

1) Let's solve in

ℝ

\ln (x - 1) = \ln (2 - x)

Since the ln function is strictly increasing,

\ln (a) = \ln (b)

is equivalent to

a = b

x - 1 = 2 - x

ⓘ We add

1

to both sides■Start from

2

and move forward

1

on the number line:

2 + 1

3

x = 3 - x

ⓘ We add

x

to both sides

x + x = 3

ⓘ Like terms are grouped by adding their coefficients:

x + x

2 x

2 x = 3

ⓘ Both sides are divided by

2

Candidates, pending the domain check:

x = \frac{3}{2}

2) Let's evaluate the domain of

\ln (x - 1) = \ln (2 - x)

For the expression

\ln (x - 1)

to be defined, we must have:

x - 1 > 0

For the expression

\ln (2 - x)

to exist, you must check:

2 - x > 0

3) Let's solve in

ℝ

x - 1 > 0

ⓘ We add

1

to both sides

Solution obtained:

x > 1

4) Let's solve in

ℝ

2 - x > 0

ⓘ We add

x

to both sides

2 > x

Solution found:

x < 2

Intersecting

x > 1

and

x < 2

gives

x \in (1, 2)

Domain constraints:

x \in (1, 2)

▶Solution:

x = \frac{3}{2}

5) The numeric value of:

x = \frac{3}{2}

equals:

▷

x = 1.5