If-ax-loga-bx-logb-then-prove-that-x-1-ab-

Question Number 196690 by MATHEMATICSAM last updated on 29/Aug/23

If {(a x)}^{\log a} = {(b x)}^{\log b} then prove that

x = \frac{1}{a b} .

Answered by BaliramKumar last updated on 29/Aug/23

loga \cdot \log (ax) = logb \cdot \log (bx)

loga \cdot (loga + logx) = logb \cdot (logb + logx)

\log^{2} a + loga \cdot logx = \log^{2} b + logb \cdot logx

loga \cdot logx - logb \cdot logx = \log^{2} b - \log^{2} a

logx (loga - logb) = (logb - loga) (logb + loga)

logx \cdot \log (\frac{a}{b}) = - \log (\frac{a}{b}) \cdot logab

logx = - \log (ab)

logx = \log (\frac{1}{ab})

x = \frac{1}{ab}