Logarithms Questions

Question Number 179722 by Acem last updated on 01/Nov/22

$$\mathrm{1000}^{\mathrm{500}\:{x}} \:=\:\sqrt{\mathrm{200}} \\$$

Commented by CElcedricjunior last updated on 02/Nov/22

$$\mathrm{1000}^{\mathrm{500}\boldsymbol{\mathrm{x}}} =\sqrt{\mathrm{200}} \\$$$$\mathrm{500}\boldsymbol{\mathrm{xln}}\mathrm{10}^{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{\mathrm{ln}}\mathrm{2}+\boldsymbol{\mathrm{ln}}\mathrm{100}\right) \\$$$$=>\boldsymbol{\mathrm{x}}=\frac{\mathrm{3}\boldsymbol{\mathrm{ln}}\mathrm{2}+\mathrm{2}\boldsymbol{\mathrm{ln}}\mathrm{5}}{\mathrm{3000}\left(\boldsymbol{\mathrm{ln}}\mathrm{2}+\boldsymbol{\mathrm{ln}}\mathrm{5}\right)}=\frac{\mathrm{1}}{\mathrm{1500}}+\frac{\boldsymbol{\mathrm{ln}}\mathrm{2}}{\mathrm{3000}\boldsymbol{\mathrm{ln}}\mathrm{10}} \\$$$$\\$$

Commented by Acem last updated on 03/Nov/22

$${Good}\:{Sir}! \\$$

Answered by Acem last updated on 01/Nov/22

$${Other}\:{way} \\$$$$\\$$$$\mathrm{10}^{\mathrm{1500}{x}} =\:\mathrm{10}\sqrt{\mathrm{2}} \\$$$$\mathrm{1500}{x}\:\mathrm{log}\:\mathrm{10}=\:\mathrm{log}\:\mathrm{10}+\:\mathrm{log}\:\mathrm{2} \\$$$${x}=\:\frac{\mathrm{1}+\mathrm{log}\:\sqrt{\mathrm{2}}}{\mathrm{1500}} \\$$