Logarithm question

Question

Log_2x-1(4.5x)=2

find domain and solution

my initial answers are x>1/2 for the domain and x=2 for the solution but it’s incorrect.

Answer 1

log_2x-1(4.5x) = 2

(2x - 1)² = 4.5x domain is ℜ

The rational root is 2.

Answer 2

Your domain is determined by the expression inside the parentheses 4.5x > 0

One approach to solving the equation is changing form

(2x-1)^2=4.5x

Distribute the square

4x^2-4x+1=4.5x

Double both sides

8x^2-8x+2=9x

Now solve the quadratic

8x^2-17x+2=0