If-b-gt-1-x-gt-0-and-2x-log-b-2-3x-log-b-3-0-then-x-is-

Question Number 111721 by Aina Samuel Temidayo last updated on 04/Sep/20

If b > 1, x > 0 and {(2 x)}^{\log_{b} 2} - {(3 x)}^{\log_{b} 3} = 0,

then x is

Answered by Her_Majesty last updated on 05/Sep/20

{(2 x)}^{{l o g}_{b} 2} = {(3 x)}^{{l o g}_{b} 3}

{l o g}_{b} 2 \times l n (2 x) = {l o g}_{b} 3 \times l n (3 x)

\frac{l n 2}{l n b} l n (2 x) = \frac{l n 3}{l n b} l n (3 x)

l n 2 (l n 2 + l n x) = l n 3 (l n 3 + l n x)

{(l n 2)}^{2} + l n 2 l n x = {(l n 3)}^{2} + l n 3 l n x

(l n 3 - l n 2) l n x = {(l n 2)}^{2} - {(l n 3)}^{2}

l n x = \frac{(l n 3 - l n 3) (l n 2 + l n 3)}{l n 3 - l n 2}

l n x = - (l n 2 + l n 3)

l n x = - l n 6

l n x = l n \frac{1}{6}

x = \frac{1}{6}

Commented by Aina Samuel Temidayo last updated on 05/Sep/20

Thanks .