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

log(√(125)) ∙ln10 ∙log_5 e=? help me

$$\mathrm{log}\sqrt{\mathrm{125}}\:\centerdot\mathrm{ln10}\:\centerdot\mathrm{log}_{\mathrm{5}} \mathrm{e}=? \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$

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

what is the practice sir?

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

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

thanks sir

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

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

(3/2)ln10 But if the base is 10 then (3/2)log_(10) 10=(3/2)

$$\frac{\mathrm{3}}{\mathrm{2}}{ln}\mathrm{10}\: \\ $$$${But}\:{if}\:{the}\:{base}\:{is}\:\mathrm{10}\:{then}\:\frac{\mathrm{3}}{\mathrm{2}}{log}_{\mathrm{10}} \mathrm{10}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Answered by 1549442205 last updated on 29/Jun/20

Applying the property of logarithm: log_a b×log_b c=log_a c we have: Log_5 e×ln10.log(√(125))=log_5 10×log_(10) (√(125)) log_5 (√(125))=log_5 5^(3/2) =(3/2) Result equal to (3/2) If log(√(125)) isn′t base 10 then equal to log_5 10×log(√(125))=(3/2)log_5 10×log5= (3/2)log_x 5×log_5 10=(3/2)log_x 10

$$\mathrm{Applying}\:\mathrm{the}\:\mathrm{property}\:\mathrm{of}\:\mathrm{logarithm}: \\ $$$$\mathrm{log}_{\mathrm{a}} \mathrm{b}×\mathrm{log}_{\mathrm{b}} \mathrm{c}=\mathrm{log}_{\mathrm{a}} \mathrm{c}\:\mathrm{we}\:\mathrm{have}: \\ $$$$\mathrm{Log}_{\mathrm{5}} \mathrm{e}×\mathrm{ln10}.\mathrm{log}\sqrt{\mathrm{125}}=\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log}_{\mathrm{10}} \sqrt{\mathrm{125}} \\ $$$$\mathrm{log}_{\mathrm{5}} \sqrt{\mathrm{125}}=\mathrm{log}_{\mathrm{5}} \mathrm{5}^{\frac{\mathrm{3}}{\mathrm{2}}} =\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{Result}}\:\boldsymbol{\mathrm{equal}}\:\boldsymbol{\mathrm{to}}\:\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{If}\:\mathrm{log}\sqrt{\mathrm{125}}\:\mathrm{isn}'\mathrm{t}\:\mathrm{base}\:\mathrm{10}\:\mathrm{then}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log}\sqrt{\mathrm{125}}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{5}} \mathrm{10}×\mathrm{log5}= \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{x}} \mathrm{5}×\mathrm{log}_{\mathrm{5}} \mathrm{10}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{x}} \mathrm{10} \\ $$

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

Sir the base is not 10

$${Sir}\:{the}\:{base}\:{is}\:{not}\:\mathrm{10} \\ $$

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