log-3-x-3-log-2-x-2-2lg6-lg2-1-find-x-

Question Number 142755 by ERA last updated on 05/Jun/21

\log_{3} x^{3} + \log_{2} x^{2} = \frac{2 \lg 6}{\lg 2} + 1 find x

Commented by iloveisrael last updated on 05/Jun/21

w h a t d i f f e r e n t \log a n d \lg ?

Commented by qaz last updated on 05/Jun/21

lgx = \log_{10} x

Answered by iloveisrael last updated on 05/Jun/21

\Rightarrow \log_{3} (x^{3}) + \log_{2} (x^{2}) = 2 \log_{2} (3 \times 2) + \log_{3} (3)

\Rightarrow 3 \log_{3} (x) + 2 \log_{2} (x) = 2 \log_{2} (2) + 3 \log_{3} (3)

\Rightarrow {3 \log}_{3} (\frac{x}{3}) = {2 \log}_{2} (\frac{2}{x})

\Rightarrow {3 \log}_{3} (\frac{x}{3}) = - {2 \log}_{2} (\frac{x}{2})

\Rightarrow \frac{3 \ln x - 3 \ln 3}{\ln 3} = \frac{- 2 \ln x + 2 \ln 2}{\ln 2}

\Rightarrow 3 \ln 2 . \ln x - 3 \ln 2 . \ln 3 = - 2 \ln 3 . \ln x + 2 \ln 2 . \ln 3

\Rightarrow 2 \ln x . \ln 8 = 2 \ln 2 . \ln 8

\Rightarrow x = 2