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Question Number 76034 by hmamarques1994@gmail.com last updated on 22/Dec/19

$$$$$${53}^{{\mathbf{log}}_{\mathbf{x}}\left(7\right)}=\sqrt{\mathbf{x}}$$$$$$$$\mathbf{x}=?$$$$$$

Answered by MJS last updated on 22/Dec/19

$${53}^{\frac{\ln 7}{\ln x}}=\sqrt{x}$$$$\frac{\ln 7}{\ln x}\ln 53=\ln \sqrt{x}$$$$\frac{\ln 7\ln 53}{\ln x}=\frac{\ln x}{2}$$$${\left(\ln x\right)}^2=2\ln 7\ln 53$$$$\ln x=\pm \sqrt{2\ln 7\ln 53}$$$$x={\mathrm{e}}^{\pm \sqrt{2\ln 7\ln 53}}$$$$x\approx .0196268\vee x\approx 50.9508$$

Commented by David Danile last updated on 04/Jan/20

$${can}^{\prime }twesolvethisusingthefacttbat$$$$$$$$b^{{log}_x{a}^c}=c?$$

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