Solve 2log25x^125log512+2log64/31

$$\frac { 2 \log 25 x ^ { 125 } \log 512 + 2 \log 64 } { 31 }$$

Factor

$$\frac{6\ln(2)\left(375\ln(x)+2\ln(1250)\right)}{31\ln(10)^{2}}$$

Solution Steps

Consider 2\ln(25x^{125})\ln(10)^{-1}\ln(512)\ln(10)^{-1}+2\ln(64)\ln(10)^{-1}. Factor out 2.

$$2\left(\log_{10}\left(25x^{125}\right)\log_{10}\left(512\right)+\log_{10}\left(64\right)\right)$$

Consider \ln(25x^{125})\ln(512)\ln(10)^{-2}+\ln(64)\times \frac{1}{\ln(10)}. Factor out \frac{1}{\ln(10)}.

$$\frac{9\ln(2)\log_{10}\left(25x^{125}\right)+6\ln(2)}{\ln(10)}$$

Rewrite the complete factored expression. Simplify.

$$\frac{2\left(9\ln(2)\log_{10}\left(25x^{125}\right)+6\ln(2)\right)}{31\ln(10)}$$

Evaluate

$$\frac{2\ln(2)\left(1125\ln(x)+6\ln(1250)\right)}{31\ln(10)^{2}}$$