Solve 2sqrt{1+lnx}*lnx-4/3(1+lnx)^frac{3{2}}

$$2 \sqrt { 1 + \ln x } \cdot \ln x - \frac { 4 } { 3 } ( 1 + \ln x ) ^ { \frac { 3 } { 2 } }$$

Factor

$$\frac{2\sqrt{\ln(x)+1}\left(\ln(x)-2\right)}{3}$$

Solution Steps

Factor out \frac{2}{3}.

$$\frac{2\left(3\sqrt{1+\log_{e}\left(x\right)}\log_{e}\left(x\right)-2\left(1+\log_{e}\left(x\right)\right)^{\frac{3}{2}}\right)}{3}$$

Consider 3\sqrt{1+\ln(x)\times \frac{1}{\ln(e)}}\ln(x)\times \frac{1}{\ln(e)}-2\left(1+\ln(x)\times \frac{1}{\ln(e)}\right)^{\frac{3}{2}}. Factor out \sqrt{1+\log_{e}\left(x\right)}.

$$\sqrt{1+\log_{e}\left(x\right)}\left(3\log_{e}\left(x\right)-2\left(1+\log_{e}\left(x\right)\right)\right)$$

Rewrite the complete factored expression.

$$\frac{2\sqrt{1+\log_{e}\left(x\right)}\left(3\log_{e}\left(x\right)-2\left(1+\log_{e}\left(x\right)\right)\right)}{3}$$

Evaluate

$$\frac{2\sqrt{\ln(x)+1}\left(\ln(x)-2\right)}{3}$$