what-is-2-log2x-3-log3x-

Question Number 103961 by MMpotulo last updated on 18/Jul/20

w h a t i s

2^{l o g 2 x} = 3^{l o g 3 x}

Answered by bramlex last updated on 18/Jul/20

\ln (2^{\ln (2 x)}) = \ln (3^{\ln (3 x)})

\ln (2 x) . \ln (2) = \ln (3 x) . \ln (3)

{\ln 2 + \ln x} . \ln 2 = {\ln 3 + \ln x} . \ln 3

\ln^{2} (2) - \ln^{2} (3) = - (\ln 2 - \ln 3) . \ln x

\ln (2) + \ln (3) = - \ln x

- \ln (6) = \ln (x) \Rightarrow x = \frac{1}{6} ★

Commented by MMpotulo last updated on 18/Jul/20

t h e r e i s a n o t h e r w a y o f d o i n g i t r i g h t

w h i c h i s

2^{l o g 2 x} = 3^{l o g 3 x}

l o g 2^{l o g 2 x} = l o g 3^{l o g 3 x}

(l o g 2 x) (l o g 2) = (l o g 3 x) (l o g 3)

(l o g 2 \times l o g

x) (l o g 2) = (l o g 3) (l o g x) (l o g 3)

? t h e n ?

Commented by bobhans last updated on 18/Jul/20

w r o n g . (\ln 2 + \ln x) . \ln 2 = (\ln 3 + \ln x) . \ln 3

Answered by Dwaipayan Shikari last updated on 18/Jul/20

l o g 2 x l o g 2 = l o g 3 x l o g 3

(l o g 2 + l o g x) l o g 2 = (l o g 3 + l o g x) l o g 3

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

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

- (l o g 6) = l o g x

{l o g}_{x} 6 = - 1

x = \frac{1}{6}

Answered by OlafThorendsen last updated on 18/Jul/20

\log (2^{\log 2 x}) = \log (3^{\log 3 x})

\log 2 x \log 2 = \log 3 x \log 3

(\log 2 + \log x) \log 2 = (\log 3 + \log x) \log 3

\log x (\log 3 - \log 2) = \log^{2} 2 - \log^{2} 3

\log x (\log 3 - \log 2) = (\log 2 - \log 3) (\log 2 + \log 3)

\log x = - (\log 2 + \log 3)

\log x = - \log 6

\log x = \log \frac{1}{6}

x = \frac{1}{6}