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Question Number 206177 by MATHEMATICSAM last updated on 08/Apr/24
Find total number of solutions of  the equation sinx = logx.
$$\mathrm{Find}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{sin}{x}\:=\:\mathrm{log}{x}. \\ $$
Answered by Frix last updated on 08/Apr/24
log x =ln x ?  −1≤ln x ≤1 ⇔ e^(−1) ≤x≤e  ⇒ one solution at x≈2.21910715    log x =log_(10)  x ?  −1≤log x ≤1 ⇔ (1/(10))≤c≤10  ⇒ three solutions  x_1 ≈2.69625656  x_2 ≈7.32834778  x_3 ≈8.26382974
$$\mathrm{log}\:{x}\:=\mathrm{ln}\:{x}\:? \\ $$$$−\mathrm{1}\leqslant\mathrm{ln}\:{x}\:\leqslant\mathrm{1}\:\Leftrightarrow\:\mathrm{e}^{−\mathrm{1}} \leqslant{x}\leqslant\mathrm{e} \\ $$$$\Rightarrow\:\mathrm{one}\:\mathrm{solution}\:\mathrm{at}\:{x}\approx\mathrm{2}.\mathrm{21910715} \\ $$$$ \\ $$$$\mathrm{log}\:{x}\:=\mathrm{log}_{\mathrm{10}} \:{x}\:? \\ $$$$−\mathrm{1}\leqslant\mathrm{log}\:{x}\:\leqslant\mathrm{1}\:\Leftrightarrow\:\frac{\mathrm{1}}{\mathrm{10}}\leqslant{c}\leqslant\mathrm{10} \\ $$$$\Rightarrow\:\mathrm{three}\:\mathrm{solutions} \\ $$$${x}_{\mathrm{1}} \approx\mathrm{2}.\mathrm{69625656} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{7}.\mathrm{32834778} \\ $$$${x}_{\mathrm{3}} \approx\mathrm{8}.\mathrm{26382974} \\ $$

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