Bài 3:$\log \frac{{{x^2} + x + 1}}{{2{x^2} - 2x + 3}} = {x^2} - 3x + 2$ $ \Leftrightarrow \log ({x^2} + x + 1) - \log (2{x^2} - 2x + 3) = (2{x^2} - 2x + 3) - ({x^2} + x + 1)$$ \Leftrightarrow \log ({x^2} + x + 1) + ({x^2} + x + 1) = \log (2{x^2} - 2x + 3) + (2{x^2} - 2x + 3)$Đặt $f(t) = \log t + t$Do ${x^2} + x + 1 > 0, 2{x^2} - 2x + 3 > 0 \forall x \in R$ nên ta xét $t \in (0; + \infty )$Ta có: $f'(t) = \frac{1}{{t\ln 10}} + 1 > 0\forall t \in (0; + \infty )$Suy ra hàm số f(t) đồng biến trên khoảng $(0; + \infty )$Suy ra: $f({x^2} + x + 1) = f(2{x^2} - 2x + 3)$ $\Leftrightarrow 2{x^2} - 2x + 3 = {x^2} + x + 1$$ \Leftrightarrow {x^2} - 3x + 2$$\Leftrightarrow x = 1 \vee x = 2$
Bài 3:$\log \frac{{{x^2} + x + 1}}{{2{x^2} - 2x + 3}} = {x^2} - 3x + 2$ $ \Leftrightarrow \log ({x^2} + x + 1) - \log (2{x^2} - 2x + 3) = (2{x^2} - 2x + 3) - ({x^2} + x + 1)$$ \Leftrightarrow \log ({x^2} + x + 1) + ({x^2} + x + 1) = \log (2{x^2} - 2x + 3) + (2{x^2} - 2x + 3)$Đặt $f(t) = \log t + t$Do ${x^2} + x + 1 > 0, 2{x^2} - 2x + 3 > 0 \forall x \in R$ nên ta xét $t \in (0; + \infty )$Ta có: $f'(t) = \frac{1}{{t\ln 10}} + 1 > 0\forall t \in (0; + \infty )$Suy ra hàm số f(t) đồng biến trên khoảng $(0; + \infty )$Suy ra: $f({x^2} + x + 1) = f(2{x^2} - 2x + 3)$ $\Leftrightarrow 2{x^2} - 2x + 3 = {x^2} + x + 1$$ \Leftrightarrow {x^2} - 3x + 2$$\Leftrightarrow x = 1 \vee x = 2$
Bài 3:$\log \frac{{{x^2} + x + 1}}{{2{x^2} - 2x + 3}} = {x^2} - 3x + 2$ $ \Leftrightarrow \log ({x^2} + x + 1) - \log (2{x^2} - 2x + 3) = (2{x^2} - 2x + 3) - ({x^2} + x + 1)$$ \Leftrightarrow \log ({x^2} + x + 1) + ({x^2} + x + 1) = \log (2{x^2} - 2x + 3) + (2{x^2} - 2x + 3)$Đặt $f(t) = \log t + t$Do ${x^2} + x + 1 > 0, 2{x^2} - 2x + 3 > 0 \forall x \in R$ nên ta xét $t \in (0; + \infty )$Ta có: $f'(t) = \frac{1}{{t\ln 10}} + 1 > 0\forall t \in (0; + \infty )$Suy ra hàm số f(t) đồng biến trên khoảng $(0; + \infty )$Suy ra: $f({x^2} + x + 1) = f(2{x^2} - 2x + 3)$ $\Leftrightarrow 2{x^2} - 2x + 3 = {x^2} + x + 1$$ \Leftrightarrow {x^2} - 3x + 2$$\Leftrightarrow x = 1 \vee x = 2$