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Fundamental Laws of Logarithms

If xa=y, yb=z...

Question

If $x^{a} = y, y^{b} = z, z^{c} = x$ then, $a b c =$ ___.

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Solution

$a b c = l o g_{x} y . l o g_{y} z . l o g_{z} x = l o g_{x} x = 1$
Alt. $x = z^{c} = (y^{b})^{c} = y^{b c} = (x^{a})^{b c} = x^{a b c}$
$∴ a b c = 1 \Rightarrow (c)$ .

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Similar questions

x^{a} = y, y^{b} = z, z^{c} = x

x^{a} = y

y^{b} = z

z^{c} = x

, then prove that

a b c = 1

\frac{x}{x a + y b + z c} = \frac{y}{y a + z b + x c} = \frac{z}{z a + x b + y c}

x + y + z \neq 0

, then each ratio is:

Q. Three vectors

¯ ¯ ¯ a, ¯ ¯ b a n d ¯ ¯ c

are co planar then for scalars

x, y, z

x ¯ ¯ ¯ a + y ¯ ¯ b + z ¯ ¯ c

∣ ∣ ∣ ∣ \begin{matrix} a & p & x b & q & y c & r & z \end{matrix} ∣ ∣ ∣ ∣ = 16

∣ ∣ ∣ ∣ \begin{matrix} p + x & a + x & a + p q + y & b + y & b + q r + z & c + z & c + r \end{matrix} ∣ ∣ ∣ ∣

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