Question Number 100832 by student work last updated on 28/Jun/20
$$\mathrm{log}\sqrt{\mathrm{125}}\:\centerdot\mathrm{ln10}\:\centerdot\mathrm{log}_{\mathrm{5}} \mathrm{e}=? \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$
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
$$\frac{\mathrm{3}}{\mathrm{2}}{ln}\mathrm{10}\: \\ $$$${But}\:{if}\:{the}\:{base}\:{is}\:\mathrm{10}\:{then}\:\frac{\mathrm{3}}{\mathrm{2}}{log}_{\mathrm{10}} \mathrm{10}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Answered by 1549442205 last updated on 29/Jun/20
$$\mathrm{Applying}\:\mathrm{the}\:\mathrm{property}\:\mathrm{of}\:\mathrm{logarithm}: \\ $$$$\mathrm{log}_{\mathrm{a}} \mathrm{b}×\mathrm{log}_{\mathrm{b}} \mathrm{c}=\mathrm{log}_{\mathrm{a}} \mathrm{c}\:\mathrm{we}\:\mathrm{have}: \\ $$$$\mathrm{Log}_{\mathrm{5}} \mathrm{e}×\mathrm{ln10}.\mathrm{log}\sqrt{\mathrm{125}}=\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log}_{\mathrm{10}} \sqrt{\mathrm{125}} \\ $$$$\mathrm{log}_{\mathrm{5}} \sqrt{\mathrm{125}}=\mathrm{log}_{\mathrm{5}} \mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} =\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{Result}}\:\boldsymbol{\mathrm{equal}}\:\boldsymbol{\mathrm{to}}\:\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{log}\sqrt{\mathrm{125}}\:\mathrm{isn}'\mathrm{t}\:\mathrm{base}\:\mathrm{10}\:\mathrm{then}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log}\sqrt{\mathrm{125}}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log5}= \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{x}} \mathrm{5}×\mathrm{log}_{\mathrm{5}} \mathrm{10}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{x}} \mathrm{10} \\ $$
Commented by Dwaipayan Shikari last updated on 28/Jun/20
$${Sir}\:{the}\:{base}\:{is}\:{not}\:\mathrm{10} \\ $$