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Question Number 63984 by Rio Michael last updated on 11/Jul/19

$$isittrue?$$$$e^{lnx}=x?$$$$ifsothen\frac{d}{dx}\left(e^{lnx}\right)=?$$

Commented by mathmax by abdo last updated on 12/Jul/19

$$yesitstrueifxrealifx=zcomplexwehavez=r{e}^{i\theta }\Rightarrow$$$$ln\left(z\right)=ln\left(r\right)+i(\theta +2k\pi )\Rightarrow e^{ln\left(z\right)}=r{e}^{i(\theta +2k\pi )}={re}^{i\theta }=z$$$$therelationistruealsoinC.$$

Commented by mr W last updated on 11/Jul/19

$$yes!e^{\ln x}=x!$$$$\frac{d\left(e^{\ln x}\right)}{dx}=e^{\ln x}\times \frac{d\left(\ln x\right)}{dx}=e^{\ln x}\times \frac{1}{x}=x\times \frac{1}{x}=1$$

Commented by Rio Michael last updated on 11/Jul/19

$$nice.$$

Commented by kaivan.ahmadi last updated on 11/Jul/19

$$y=e^{lnx}\Rightarrow lny=lnxlne=lnx\Rightarrow$$$$lny-lnx=0\Rightarrow ln\frac{y}{x}=0\Rightarrow \frac{y}{x}=e^0=1\Rightarrow y=x$$

Commented by kaivan.ahmadi last updated on 11/Jul/19

$$generally$$$$a^{{log}_{a}x}=xsince$$$$ify=a^{{log}_{a}x}\Rightarrow {log}_{a}y={log}_a{xlog}_{a}a={log}_{a}x\Rightarrow$$$${log}_{a}y-{log}_{a}x=0\Rightarrow {log}_a\frac{y}{x}=0\Rightarrow$$$$\frac{y}{x}=a^0=1\Rightarrow y=x$$

Answered by MJS last updated on 12/Jul/19

$$a+b=c\iff a=c-b\iff b=c-a$$$$\mathrm{we}\mathrm{need}\mathrm{only}\mathrm{one}\mathrm{kind}\mathrm{of}\mathrm{operation}$$$$\mathrm{same}\mathrm{here}:$$$$a\times b=c\iff a=\frac{c}{b}\iff b=\frac{c}{a}$$$${\mathrm{that}}^{\prime }\mathrm{s}\mathrm{because}\mathrm{addition}\mathrm{and}\mathrm{multiplication}$$$$\mathrm{both}\mathrm{are}\mathrm{commutative}$$$$a+b=b+c$$$$a\times b=b\times a$$$$$$$$\mathrm{but}$$$$a^b\ne b^a\mathrm{in}\mathrm{general}$$$$a^b=c\iff a=\sqrt[b]{c}\iff b={\log }_{a}c$$$$a^b=?\mathrm{with}a=\sqrt[b]{c}$$$${\left(\sqrt[b]{c}\right)}^b=c$$$$a^b=?\mathrm{with}b={\log }_{a}c$$$$a^{{\log }_{a}c}=c$$$$$$$$\ln x={\log }_{\mathrm{e}}x\mathrm{but}\mathrm{some}\mathrm{people}\mathrm{use}\log x\mathrm{instead}$$$$\mathrm{we}\mathrm{should}\mathrm{not}\mathrm{because}\mathrm{usually}\log x\mathrm{means}$$$${\log }_{10}x$$$$$$$$\mathrm{btw}$$$${\log }_{a}b=\frac{\ln b}{\ln a}$$

Commented by Rio Michael last updated on 12/Jul/19

$$thankssomuch$$

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