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Question Number 161362 by cortano last updated on 17/Dec/21

$$\log {}_{\sqrt{\frac{x}{3}}}{(3x-54)}^{\log {}_3\left(x\right)}=18-3\log {}_{\frac{x}{3}}\left(x^2\right)$$$$x=?$$

Commented by bobhans last updated on 17/Dec/21

$$\frac{\log {}_3\left(\mathrm{x}\right)[1+\log {}_3(\mathrm{x}-18)]}{\frac{1}{2}[\log {}_3\left(\mathrm{x}\right)-1]}=18-\frac{6\log {}_3\left(\mathrm{x}\right)}{\log {}_3\left(\mathrm{x}\right)-1}$$$$\frac{2\log {}_3\left(\mathrm{x}\right)[1+\log {}_3(\mathrm{x}-18)]}{\log {}_3\left(\mathrm{x}\right)-1}=\frac{12\log {}_3\left(\mathrm{x}\right)-18}{\log {}_3\left(\mathrm{x}\right)-1}$$$$2\log {}_3\left(\mathrm{x}\right)+2\log {}_3\left(\mathrm{x}\right)\log {}_3(\mathrm{x}-18)=12\log {}_3\left(\mathrm{x}\right)-18$$$$\log {}_3\left(\mathrm{x}\right)\log {}_3(\mathrm{x}-18)-5\log {}_3\left(\mathrm{x}\right)+9=0$$$$$$

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