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solve-log-7-2-log-49-x-log-7-3-




Question Number 171990 by Mikenice last updated on 22/Jun/22
solve:  log_7 2 + log_(49) x =log_7 (√3)
$${solve}: \\ $$$${log}_{\mathrm{7}} \mathrm{2}\:+\:{log}_{\mathrm{49}} {x}\:={log}_{\mathrm{7}} \sqrt{\mathrm{3}} \\ $$
Answered by nurtani last updated on 23/Jun/22
⇔ log_7  2+log_7  x^(1/2) =log_7  3^(1/2)   ⇔ log_7  2x^(1/2) = log_7  3^(1/2)   ⇔ 2x^(1/2) = 3^(1/2)   ⇔ (2x^(1/2) )^2 = (3^(1/2) )^2   ⇔ 4x = 3  ⇔ x = (3/4)
$$\Leftrightarrow\:{log}_{\mathrm{7}} \:\mathrm{2}+{log}_{\mathrm{7}} \:{x}^{\frac{\mathrm{1}}{\mathrm{2}}} ={log}_{\mathrm{7}} \:\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\Leftrightarrow\:{log}_{\mathrm{7}} \:\mathrm{2}{x}^{\frac{\mathrm{1}}{\mathrm{2}}} =\:{log}_{\mathrm{7}} \:\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\Leftrightarrow\:\mathrm{2}{x}^{\frac{\mathrm{1}}{\mathrm{2}}} =\:\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\Leftrightarrow\:\left(\mathrm{2}{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{2}} =\:\left(\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\Leftrightarrow\:\mathrm{4}{x}\:=\:\mathrm{3} \\ $$$$\Leftrightarrow\:{x}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

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