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Algebra> If a,b,c are distinct positive real nos s...

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If a,b,c are distinct positive real nos such that log_ba.log_ca-log_aa + log_ab.log_cb-log_bb + log_ac.log_bc-log_cc = 0,then prove that abc=1.

Anusuya Sahoo , 8 Years ago

Grade 9

1 Answers

gayatri

Last Activity: 8 Years ago

log_aa = log_bb = log_cc = 1 and log_ba = $\frac{log a}{log b}$

$\frac{(loga)^{2}}{logb \times logc } - 1$ $+ \frac{(logb)^{2}}{loga \times logc } -1$ $+ \frac{(logc)^{2}}{loga \times logb } - 1 = 0$

By making the denominator common and bringing 3 to the LHS of the equation :

$\frac{(loga)^{3}}{loga \times logb \times logc} + \frac{(logb)^{3}}{loga \times logb \times logc} + \frac{(logc)^{3}}{loga \times logb \times logc} = 3$

$(loga)^{3} + (logb)^{3} + \(logc)^{3} = 3 \times loga\times logb\times logc$

$\Rightarrow loga + logb + logc =0$ $\textbf{}\textbf{[} a^{3} + b^{3}+c^{3} = 3 abc \Rightarrow a+b+c = 0 \textbf{]}}$

$\Rightarrow log (abc)=0$ $\textbf{[ loga + logb + log c = log (abc)]}$

$\Rightarrow abc= 1$

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