Solve log(2x/sqrt{2})+3log(2^x)+log({left(1/2right)}^x)

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

Factor

$$\frac{\log_{10}\left(e\right)\left(2\ln(x)+4\ln(2)x+\ln(2)\right)}{2}$$

Solution Steps

Rationalize the denominator of \frac{2x}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\log_{10}\left(\frac{2x\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)+3\log_{10}\left(2^{x}\right)+\log_{10}\left(\left(\frac{1}{2}\right)^{x}\right))$$

The square of \sqrt{2} is 2.

$$factor(\log_{10}\left(\frac{2x\sqrt{2}}{2}\right)+3\log_{10}\left(2^{x}\right)+\log_{10}\left(\left(\frac{1}{2}\right)^{x}\right))$$

Cancel out 2 and 2.

$$factor(\log_{10}\left(x\sqrt{2}\right)+3\log_{10}\left(2^{x}\right)+\log_{10}\left(\left(\frac{1}{2}\right)^{x}\right))$$

Factor out \frac{1}{\ln(10)}.

$$\frac{\ln(x\sqrt{2})+3\ln(2^{x})+\ln(\left(\frac{1}{2}\right)^{x})}{\ln(10)}$$

Evaluate

$$\frac{\log(e)\left(2\ln(x)+4\ln(2)x+\ln(2)\right)}{2}$$