log_yx log_zx log_xx = log_x log_x/log_ylog_z log_x/log_x = (log_x)^3 log_xlog_ylog_z/log_xlog_ylog_z = (logx)^2 log_ylog_z/log_ylog_z ...

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Implicit Differentiation

logyx.logzx-l...

Question

$(l o g_{y} x . l o g_{z} x - l o g_{x} x) = ?$

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Solution

$l o g_{y} x l o g_{z} x - l o g_{x} x$

$= \frac{l o g_{x} l o g_{x}}{l o g_{y} l o g_{z}} - \frac{l o g_{x}}{l o g_{x}}$

$= \frac{(l o g_{x})^{3} - l o g_{x} l o g_{y} l o g_{z}}{l o g_{x} l o g_{y} l o g_{z}}$

$= \frac{(l o g x)^{2} - l o g_{y} l o g_{z}}{l o g_{y} l o g_{z}}$

$= \frac{l o g \frac{x^{l o g x}}{z^{l o g y}}}{l o g z^{l o g y}}$

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