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Category: Logarithms

64-8pi-4pi-8pi-4-tan-1-2-4tan-1-1-2-2-8-pi-2-64-16pi-8pi-16-tan-1-2-4tan-1-1-2-2-3-2-2-32tan-1-4-8-6-4-32sin-tan-1-4-8-6-4-pi-4-tan-1-2-4tan-1-1-2-

Question Number 227232 by fantastic2 last updated on 07/Jan/26 $$\left(\mathrm{64}−\left(\mathrm{8}\pi+\mathrm{4}\pi\right)\right)−\left(\mathrm{8}\pi−\left(\mathrm{4}\left(\mathrm{tan}^{−\mathrm{1}} \mathrm{2}+\mathrm{4tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\right)+\mathrm{8}\left(\pi−\mathrm{2}\right)\right)\right)−\left(\mathrm{64}−\left(\mathrm{16}\pi+\mathrm{8}\pi−\mathrm{16}\left(\mathrm{tan}^{−\mathrm{1}} \mathrm{2}+\mathrm{4tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\right)\right)\right)−\left(\mathrm{3}.\mathrm{2}+\mathrm{2}+\left(\mathrm{32tan}^{−\mathrm{1}} \frac{\mathrm{4}.\mathrm{8}}{\mathrm{6}.\mathrm{4}}−\mathrm{32sin}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{4}.\mathrm{8}}{\mathrm{6}.\mathrm{4}}\right)−\pi\right)+\mathrm{4}\left(\mathrm{tan}^{−\mathrm{1}} \mathrm{2}+\mathrm{4tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\right) \\ $$ Terms of Service Privacy Policy…

If-log-x-y-z-log-y-z-x-log-z-x-y-prove-xyz-1-

Question Number 223512 by fantastic last updated on 27/Jul/25 $${If}\:\frac{\mathrm{log}\:{x}}{{y}−{z}}=\frac{\mathrm{log}\:{y}}{{z}−{x}}=\frac{\mathrm{log}\:{z}}{{x}−{y}} \\ $$$${prove}\:{xyz}=\mathrm{1} \\ $$ Answered by mr W last updated on 27/Jul/25 $$\frac{\mathrm{log}\:{x}}{{y}−{z}}=\frac{\mathrm{log}\:{y}}{{z}−{x}}=\frac{\mathrm{log}\:{z}}{{x}−{y}}=\frac{\mathrm{1}}{{k}}\:\:\left({k}\neq\mathrm{0}\right) \\ $$$${y}−{z}={k}\:\mathrm{log}\:{x}…

Question-222697

Question Number 222697 by gabthemathguy25 last updated on 05/Jul/25 Answered by gregori last updated on 05/Jul/25 $$\:\sqrt[{\mathrm{3}}]{{e}^{\mathrm{2ln}\:\left(\mathrm{81}\right)} }\:=\:\left(\mathrm{81}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)=\:\mathrm{9}^{\frac{\mathrm{4}}{\mathrm{3}}} \\ $$$$\:\sqrt[{\mathrm{4}}]{\left(\mathrm{3}^{\mathrm{log}\:_{\mathrm{9}} \left(\mathrm{9}^{\mathrm{2}} \right)} \right)^{\mathrm{2}} }\:=\:\sqrt{\mathrm{9}}=\:\mathrm{9}^{\frac{\mathrm{1}}{\mathrm{2}}}…

If-x-x-1-10-x-then-1-log-2-x-1-log-3-x-1-log-4-x-1-log-10-x-

Question Number 222679 by fantastic last updated on 04/Jul/25 $${If}\:{x}=\underset{{x}=\mathrm{1}} {\overset{\mathrm{10}} {\prod}}{x}\:{then}\:\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{2}} {x}}+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{3}} {x}}+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{4}} {x}}…+\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{10}} {x}}=?? \\ $$ Answered by Tawa11 last updated on 04/Jul/25…