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Question Number 100832 by student work last updated on 28/Jun/20

$$\log \sqrt{125}\cdot \ln 10\cdot {\log }_5\mathrm{e}=?$$$$\mathrm{help}\mathrm{me}$$

Commented by student work last updated on 28/Jun/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{practice}\mathrm{sir}?$$

Commented by student work last updated on 28/Jun/20

$$\mathrm{thanks}\mathrm{sir}$$

Commented by Dwaipayan Shikari last updated on 29/Jun/20

$$\frac{3}{2}ln10$$$$Butifthebaseis10then\frac{3}{2}{log}_{10}10=\frac{3}{2}$$

Answered by 1549442205 last updated on 29/Jun/20

$$\mathrm{Applying}\mathrm{the}\mathrm{property}\mathrm{of}\mathrm{logarithm}:$$$${\log }_{\mathrm{a}}\mathrm{b}\times {\log }_{\mathrm{b}}\mathrm{c}={\log }_{\mathrm{a}}\mathrm{c}\mathrm{we}\mathrm{have}:$$$${\mathrm{Log}}_5\mathrm{e}\times \ln 10.\log \sqrt{125}={\log }_{5}10\times {\log }_{10}\sqrt{125}$$$${\log }_5\sqrt{125}={\log }_5{5}^{\frac{3}{2}}=\frac{3}{2}$$$$\mathbf{Result}\mathbf{equal}\mathbf{to}\frac{3}{2}$$$$\mathrm{If}\log \sqrt{125}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{base}10\mathrm{then}\mathrm{equal}\mathrm{to}$$$${\log }_{5}10\times \log \sqrt{125}=\frac{3}{2}{\log }_{5}10\times \log 5=$$$$\frac{3}{2}{\log }_{\mathrm{x}}5\times {\log }_{5}10=\frac{3}{2}{\log }_{\mathrm{x}}10$$

Commented by Dwaipayan Shikari last updated on 28/Jun/20

$$Sirthebaseisnot10$$

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