Question

Encontre as soluções de x para a seguinte equação:


$\large\begin{array}\mathsf{\mathsf{\dfrac{x}{14}=\Big(\dfrac{2}{7}\Big)^\big{\mathsf{\ell og_x (4)}}}\end{array}$

Answer 1

$\dfrac{x}{14}=\Big (\dfrac{2}{7}\Big)^\big{\log_x(4)}$

$\log_{\dfrac{2}{7}}\dfrac{x}{14}=\log_x4$

$\dfrac{\log{\dfrac{x}{14}}}{\log{\dfrac{2}{7}}}=\dfrac{\log4}{\log{x}}$

$\log{x}\cdot\log{\dfrac{x}{14}}=\log4\cdot\log{\dfrac{2}{7}}$

$\log{x}\cdot(\log{x}-\log{14})=\log4\cdot\log{\dfrac{2}{7}}$

$\log^2{x}-\log{14}\cdot\log{x}=\log4\cdot\log{\dfrac{2}{7}}$

$\boxed{\mathsf{\log{x}\Rightarrow y}}$

$y^2-\log{14}y=\log4\cdot\log{\dfrac{2}{7}}$

$y^2-\log{14}y-\log4\log{\dfrac{2}{7}}=0$

$\boxed{a=1}\\\\\boxed{b=-\log{14}}\\\\\boxed{c=-\log4\log{\dfrac{2}{7}}}$

$\Delta=b^2-4ac$

$\Delta=(\log{14})^2-4\cdot1\cdot(-\log4\log{\dfrac{2}{7}})$

$\Delta=(\log2+\log7)^2+4\log4(\log2-\log7)$

$\Delta=\log^22+2\log2\log7+\log^27+8\log2(\log2-\log7)$

$\Delta=\log^22+2\log2\log7+\log^27+8\log^22-8\log2\log7$

$\Delta=9\log^22-6\log2\log7+\log^27$

$y=\dfrac{-b\pm\sqrt\Delta}{2a}$

$y=\dfrac{\log{14}\pm\sqrt{9\log^22-6\log2\log7+\log^27}}{2}$

$y=\dfrac{\log{14}\pm(3\log2-\log7)}{2}$

Calculando o primeiro valor:

$y_1=\dfrac{\log{14}+(3\log2-\log7)}{2}$

$y_1=\dfrac{\log7+\log2+3\log2-\log7}{2}$

$y_1=\dfrac{4\log2}{2}$

$y_1=2\log2$

$\boxed{\boxed{y_1=\log4}}$

Calculando o segundo valor:

$y_2=\dfrac{\log{14}-(3\log2-\log7)}{2}$

$y_2=\dfrac{\log7+\log2-3\log2+\log7}{2}$

$y_2=\dfrac{2\log7-2\log2}{2}$

$y_2=\log7-\log2$

$\boxed{\boxed{y_2=\log{\dfrac{7}{2}}}}$

Mas, não esqueça que:

$\boxed{\mathsf{\log{x}=y}}$

Para o primeiro valor:

$\log{x}=\log4$

Cortando log:

$\boxed{\boxed{\mathsf{x_1=4}}}$

Para o segundo valor:

$\log{x}=\log{\dfrac{7}{2}}$

Cortando log:

$\boxed{\boxed{\mathsf{x_2=\dfrac{7}{2}}}}$