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Basic of Logarithms
medium
જો $\log x:\log y:\log z = (y - z)\,:\,(z - x):(x - y)$ તો
A
${x^y}.{y^z}.{z^x} = 1$
B
${x^x}{y^y}{z^z} = 1$
C
$\root x \of x \,\root y \of y \root z \of z = 1$
D
એકપણ નહીં
Solution
(b) $\log x:\log y:\log z = y – z:z – x:x – y$
$ \Rightarrow $ ${{\log x} \over {y – z}} = {{\log y} \over {z – x}} = {{\log z} \over {x – y}} = k\,{\rm{(say)}}$
$ \Rightarrow $ $\log x = k(y – z),\,\log y = k(z – x),\,\log z = k(x – y)$
$\therefore \log x + \log y + \log z = 0$$ \Rightarrow $$x + y = 1$ $ \Rightarrow $$xyz = 1$.
$x\log x + y\log y + z\log z$
=$x.k.(y – z) + y.k.(z – x) + z.k(x – y) = 0$
$ \Rightarrow $$\log ({x^x}.{y^y}.{z^z}) = \log 1$
$\therefore x^xy^yz^z=1 $.
Standard 11
Mathematics
