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Question Number 129088 by Gulnoza last updated on 12/Jan/21

Answered by MJS_new last updated on 12/Jan/21

$$4^{{2\log }_{4}x}={\left(4^{{\log }_{4}x}\right)}^2=x^2=25$$$$\Rightarrow x=\pm 5$$

Commented by hknkrc46 last updated on 12/Jan/21

$$\log {}_{\mathit{m}}\mathit{n}=\{\begin{array}{l}\mathit{m}\in {\mathbb{R}}^{+}\mathrm{\setminus }\left\{1\right\}\\ \mathit{n}\in {\mathbb{R}}^{+}\end{array}\Rightarrow \{\begin{array}{l}\mathit{x}=5\\ \mathit{x}\ne -5\end{array}$$

Commented by MJS_new last updated on 12/Jan/21

$$x\in \mathbb{C}$$$$\ln (-5)=\ln 5+\mathrm{i}\pi$$$$4^y={\mathrm{e}}^{y\ln 4}=$$$$[y=2\frac{\ln x}{\ln 4}]$$$$={\mathrm{e}}^{2\ln x}=$$$$[x=-5]$$$$={\mathrm{e}}^{2\ln 5+2\pi \mathrm{i}}={\mathrm{e}}^{2\ln 5}{\mathrm{e}}^{2\pi \mathrm{i}}=$$$$[{\mathrm{e}}^{2\pi \mathrm{i}}=1]$$$$={\mathrm{e}}^{2\ln 5}={\left({\mathrm{e}}^{\ln 5}\right)}^2=5^2=25$$

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