Find $x$ and $y$, if $\dfrac{\log{x}}{\log{5}}$ $\,=\,$ $\dfrac{\log{36}}{\log{6}}$ $\,=\,$ $\dfrac{\log{64}}{\log{y}}$

$(1).\,\,$ $\dfrac{\log{x}}{\log{5}}$ $\,=\,$ $\dfrac{\log{36}}{\log{6}}$

$(2).\,\,$ $\dfrac{\log{64}}{\log{y}}$ $\,=\,$ $\dfrac{\log{36}}{\log{6}}$

Evaluate the Arithmetic Log expression

Firstly, let us find the value of the quotient of common logarithm of thirty-six divided by common logarithm of six by simplification. The numbers inside the logarithm in the numerator and denominator are thirty-six and six respectively. Arithmetically, the numerator thirty-six can be written as the factors of the number six.

$\implies$ $\dfrac{\log{36}}{\log{6}}$ $\,=\,$ $\dfrac{\log{(6 \times 6)}}{\log{6}}$

$=\,\,$ $\dfrac{\log{(6^2)}}{\log{6}}$

$=\,\,$ $\dfrac{\log{6^2}}{\log{6}}$

$=\,\,$ $\dfrac{2 \times \log{6}}{\log{6}}$

$=\,\,$ $2 \times \dfrac{\log{6}}{\log{6}}$

$=\,\,$ $2 \times \dfrac{\cancel{\log{6}}}{\cancel{\log{6}}}$

$=\,\,$ $2 \times 1$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{\log{36}}{\log{6}}$ $\,=\,$ $2$

Solve the Logarithmic equation to find x

$\dfrac{\log{x}}{\log{5}}$ $\,=\,$ $\dfrac{\log{36}}{\log{6}}$

$\implies$ $\dfrac{\log{x}}{\log{5}}$ $\,=\,$ $2$

$\implies$ $\log{x}$ $\,=\,$ $2 \times \log{5}$

$\implies$ $\log{x}$ $\,=\,$ $\log{5^2}$

$\implies$ $\log{x}$ $\,=\,$ $\log{25}$

$\,\,\,\therefore\,\,\,\,\,\,$ $x$ $\,=\,$ $25$

Solve the Logarithmic equation to find y

$\dfrac{\log{64}}{\log{y}}$ $\,=\,$ $\dfrac{\log{36}}{\log{6}}$

$\implies$ $\dfrac{\log{64}}{\log{y}}$ $\,=\,$ $2$

$\implies$ $\log{64}$ $\,=\,$ $2 \times \log{y}$

$\implies$ $\log{64}$ $\,=\,$ $\log{y^2}$

$\implies$ $\log{y^2}$ $\,=\,$ $\log{64}$

$\implies$ $y^2$ $\,=\,$ $64$

$\implies$ $y$ $\,=\,$ $\pm \sqrt{64}$

$\implies$ $y$ $\,=\,$ $\pm \sqrt{8^2}$

$\implies$ $y$ $\,=\,$ $\pm 8$

$\implies$ $y$ $\,=\,$ $\pm 8$

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