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Question Number 178341 by haladu last updated on 15/Oct/22

Answered by mahdipoor last updated on 15/Oct/22

log_y x=((lnx)/(lny))⇒log_b a+log_c a+log_d a=  ln(a).((1/(ln(b)))+(1/(ln(c)))+(1/(ln(d))))=q.ln(a)  ∫ ((sin(q.lna))/a^2 )da=e^(p^2 /q^2 ) .(1/( (√(1+q^2 )))).((sin(q.lnb))/(−b))  p=cos^(−1) ((1/( (√(1+q^2 )))))  b=a.e^(p/q)   −−−−−−−−−−−−  ((1/( (√(1+m^2 )))).((sin(m.lnb))/(−b)))′=  (((1/( (√(1+m^2 ))))sin(m.lnb))/b^2 )−(((m/( (√(1+m^2 )))).cos(m.lnb))/b^2 )  =((cos(n).sin(m.lnb)−sin(n).cos(m.lnb))/b^2 )  =((sin(m.lnb−n))/b^2 )                cos(n)=(1/( (√(1+m^2 ))))  =((sin(m.lna))/a^2 ).e^(−n^2 /m^2 )                b=a.e^(n/m)

$${log}_{{y}} {x}=\frac{{lnx}}{{lny}}\Rightarrow{log}_{{b}} {a}+{log}_{{c}} {a}+{log}_{{d}} {a}= \\ $$$${ln}\left({a}\right).\left(\frac{\mathrm{1}}{{ln}\left({b}\right)}+\frac{\mathrm{1}}{{ln}\left({c}\right)}+\frac{\mathrm{1}}{{ln}\left({d}\right)}\right)={q}.{ln}\left({a}\right) \\ $$$$\int\:\frac{{sin}\left({q}.{lna}\right)}{{a}^{\mathrm{2}} }{da}={e}^{{p}^{\mathrm{2}} /{q}^{\mathrm{2}} } .\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{q}^{\mathrm{2}} }}.\frac{{sin}\left({q}.{lnb}\right)}{−{b}} \\ $$$${p}={cos}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{q}^{\mathrm{2}} }}\right) \\ $$$${b}={a}.{e}^{{p}/{q}} \\ $$$$−−−−−−−−−−−− \\ $$$$\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}.\frac{{sin}\left({m}.{lnb}\right)}{−{b}}\right)'= \\ $$$$\frac{\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}{sin}\left({m}.{lnb}\right)}{{b}^{\mathrm{2}} }−\frac{\frac{{m}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}.{cos}\left({m}.{lnb}\right)}{{b}^{\mathrm{2}} } \\ $$$$=\frac{{cos}\left({n}\right).{sin}\left({m}.{lnb}\right)−{sin}\left({n}\right).{cos}\left({m}.{lnb}\right)}{{b}^{\mathrm{2}} } \\ $$$$=\frac{{sin}\left({m}.{lnb}−{n}\right)}{{b}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{cos}\left({n}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }} \\ $$$$=\frac{{sin}\left({m}.{lna}\right)}{{a}^{\mathrm{2}} }.{e}^{−{n}^{\mathrm{2}} /{m}^{\mathrm{2}} } \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}={a}.{e}^{{n}/{m}} \\ $$

Commented by haladu last updated on 16/Oct/22

  Awasome Solution!

$$\:\:\boldsymbol{\mathrm{Awasome}}\:\boldsymbol{\mathrm{Solution}}! \\ $$

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