0\ \wedge\ 2x-3 > 0\ \wedge\ x > 0\\x > 2\ \wedge\ x > 1.5\ \wedge\ x > 0\\x\in(2;\ \infty)"> l n ( x − 2 ) + l n ( 2 x − 3 ) = 2 l n x D : x − 2 > 0 ∧ 2 x − 3 > 0 ∧ x > 0 x > 2 ∧ x > 1.5 ∧ x > 0 x ∈ ( 2 ; ∞ )
l n [( x − 2 ) ( 2 x − 3 )] = l n x 2 l n ( 2 x 2 − 3 x − 4 x + 6 ) = l n x 2 l n ( 2 x 2 − 7 x + 6 ) = l n x 2 ⟺ 2 x 2 − 7 x + 6 = x 2 2 x 2 − x 2 − 7 x + 6 = 0 x 2 − 7 x + 6 = 0 x 2 − x − 6 x + 6 = 0 x ( x − 1 ) − 6 ( x − 1 ) = 0 ( x − 1 ) ( x − 6 ) = 0 ⟺ x − 1 = 0 ∨ x − 6 = 0 x = 1 ∈ / D ; x = 6 ∈ D S o l u t i o n : x = 6.
[\ln(x-2)+\ln(2x-3)=2\ln x \ D:x-2>0 \wedge 2x-3 >0 \wedge x>0\ D:x>2 \wedge 2x>3 \wedge x>0\ D:x>2 \wedge x>\frac{3}{2}\ D:x>2\ \ln(x-2)(2x-3)=\ln x^2\ 2x^2-3x-4x+6=x^2\ x^2-7x+6=0\ x^2-x-6x+6=0\ x(x-1)-6(x-1)=0\ (x-6)(x-1)=0\ x=6 \vee x=1\ 1\not \in D\Rightarrow x=6
]
The solution to the equation ln ( x − 2 ) + ln ( 2 x − 3 ) = 2 ln x is x = 6 after simplifying and verifying the domain of the logarithmic expressions. The steps involve using properties of logarithms to combine and then exponentiate the terms. Checking the domain reveals that x = 1 is not an acceptable solution.
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