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Log4x(1/2) +logx(8) +log1/2x(1/2) =1/4 |
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Answer» we know \( \log_a x=\frac{1}{\log_x a} \). so we reverse the sides of the equation: \( \log_{\frac{1}{2}} 4x+\log_8 x+\log_{\frac{1}{2}} \frac{1}{2}x=4 \) also we know \( \log_a xy=\log_a x+\log_a y\) , \( \log_a x^n=n \log_a x \) , \( \log_{a^n} x=\frac{1}{n}\log_a x \) and \( \log_a a=1 \). so we have: \( \log_{\frac{1}{2}}4+ \log_{\frac{1}{2}}x+\log_8 x+\log_{\frac{1}{2}} \frac{1}{2}+\log_{\frac{1}{2}} x=4 \) \( -2\log_2 2-\log_2 x+\frac{1}{3}\log_2 x+1-\log_2 x=4 \) \( -2+1-\frac{5}{3}\log_2 x=4 \) \( \frac{5}{3} \log_2 x=-5 \) \( \log_2 x=-3 \) \( x=2^{-3}=(\frac{1}{2})^3=\frac{1}{8} \) |
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