Verify if the logarithmic equation has real solutions ? ln(x^2-1)-ln(x-1)=ln(4)

We'll start by imposing constraints for the existence of
the logarithmic functions:

x^2 - 1 >
0

x - 1 > 0

x >
1

The common interval of values of x that make the
logarithmic functions to exist is (1 , +infinite).

We'll
solve the equation applying the quotient property to the left
side:

ln (x^2 - 1)/(x-1) = ln
4

We'll re-write the difference of 2 squares from
numerator:

x^2 - 1 =
(x-1)(x+1)

ln (x-1)(x+1)/(x-1) = ln
4

We'll simplify and we'll
get:

ln (x + 1) = ln 4

Since
the bases are matching, we'll apply one to one rule:

x + 1
= 4

x = 4 - 1

x =
3

Since the value of x belongs to the range
of admissible values, we'll accept x = 3 as a solution of the
equation.