Question

If log x =(1/2) log y = (1/5) log z, the value of x4y3z-2 i

Answer 1

Answer:

$\red {Value \: of \: x^{4} y^{3} z^{-2}} \green{= 1}$

Step-by-step explanation:

$Given \: log\:x = \frac{1}{2} \: log\:y = \frac{1}{5} \: log\:z$

$i ) log\:x = \frac{1}{2} \: log\:y$

$\implies 2log\:x = log\:y$

$\implies log \:x^{2} = log \:y$

$\boxed { \pink { nlog \:a = log \:a^{n}}}$

$\implies x^{2} = y \: ----(1)$

$ii ) log\:x = \frac{1}{5} \: log\:z$

$\implies 5log\:x = log\:z$

$\implies log \:x^{5} = log \:z$

$\boxed { \pink { nlog \:a = log \:a^{n}}}$

$\implies x^{5} = z \: ----(2)$

$Now, \: Value \: of \: x^{4} y^{3} z^{-2}\\= x^{4} \times (x^{2})^{3} \times \left( x^{5}\right)^{-2}$

$\orange { [ From \: (1) \: and \: (2) ]}$

$= x^{4} \times x^{6} \times x^{5\times (-2)}$

$\boxed {\pink { (a^{m})^{n} = a^{mn}}}$

$= x^{4+6-10}$

$\boxed { \pink { x^{m} \times x^{n} = x^{m+n}}}$

$= x^{0}\\=1$

$\boxed {\pink { a^{0} = 1 }}$

Therefore.,

$\red {Value \: of \: x^{4} y^{3} z^{-2}} \green{= 1}$

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