x-12-log-3-x-x-18-log-2-x-find-x-

Question Number 92179 by otchereabdullai@gmail.com last updated on 05/May/20

{(\frac{x}{12})}^{\log_{\sqrt{3}} x} = {(\frac{x}{18})}^{\log_{\sqrt{2}} x}

find x

Commented by john santu last updated on 05/May/20

{(x)}^{\log_{\sqrt{3}} (\frac{x}{12})} = {(x)}^{\log_{\sqrt{2}} (\frac{x}{18})}

(x - 1) (\log_{\sqrt{3}} (\frac{x}{12}) - \log_{\sqrt{2}} (\frac{x}{18})) = 0

(i) x = 1

(i i) 2 \log_{3} (\frac{x}{12}) = 2 \log_{2} (\frac{x}{18})

\log_{3} (x) - \log_{3} (12) =

\log_{2} (x) - \log_{2} (18)

\log_{3} (x) - \log_{2} (x) = \log_{3} (12) - \log_{2} (18)

\log_{3} (x) {1 - \log_{2} (3)} =

\log_{3} (6) + \log_{3} (2) - \log_{2} (6) - \log_{2} (3)

\log_{3} (x) {1 - \log_{2} (3)} = 2 . {1 - \log_{2} (3)}

\Rightarrow x = 3^{2} = 9

s o l u t i o n x = 1 \land x = 9

Commented by jagoll last updated on 05/May/20

good

Commented by otchereabdullai@gmail.com last updated on 05/May/20

thanks for your time sir!

Commented by otchereabdullai@gmail.com last updated on 05/May/20

but sir pls x = 9 do not satisfy

Commented by otchereabdullai@gmail.com last updated on 05/May/20

and please the base in the question

was \sqrt{3} and \sqrt{2} in (ii) but pls why base 3 and

2 please i want to understand thanks

sir

Commented by otchereabdullai@gmail.com last updated on 05/May/20

am much greatful sir God bless you!

Commented by john santu last updated on 05/May/20

{(9)}^{\log_{\sqrt{3}} (\frac{9}{12})} = {(9)}^{\log_{3} {(\frac{3}{4})}^{2}}

{(3)}^{\log_{3} {(\frac{3}{4})}^{4}} = \frac{81}{256} (Lhs)

Rhs {(9)}^{\log_{\sqrt{2}} (\frac{9}{18})} = {(9)}^{\log_{2} (\frac{1}{4})}

= {(3)}^{\log_{2} (\frac{1}{16})} = \frac{1}{81}

Lhs \neq Rhs

x = 1 only solution

Commented by john santu last updated on 05/May/20

we know property of logarithm

a^{\log_{b} (c)} = c^{\log_{b} (a)}

Commented by john santu last updated on 05/May/20

\log_{a^{n}} (b) = \frac{1}{n} \log_{a} (b)

\log_{3^{\frac{1}{2}}} (x) = 2 \log_{3} (x)

Commented by otchereabdullai@gmail.com last updated on 05/May/20

Am much greatful thank you sair!

please sir my final question is on how

you got the (x - 1)

Commented by jagoll last updated on 05/May/20

sir if {(f (x))}^{g (x)} = ({(f (x))}^{h (x)}

then (f (x) - 1) (g (x) - h (x)) = 0

Commented by otchereabdullai@gmail.com last updated on 05/May/20

a have really enjoy your lesson

God richly bless you