If-log-4-x-log-4-x-2-log-4-x-3-log-4-x-4-1-find-the-value-of-x-

Question Number 117392 by ZiYangLee last updated on 11/Oct/20

If \log_{4} x + {(\log_{4} x)}^{2} + {(\log_{4} x)}^{3} + {(\log_{4} x)}^{4} + \dots = 1

find the value of x .

Answered by Dwaipayan Shikari last updated on 11/Oct/20

{l o g}_{4} x + {({l o g}_{4} x)}^{2} + \dots = 1

\frac{{l o g}_{4} x}{1 - {l o g}_{4} x} = 1

\frac{{l o g}_{4} x}{{l o g}_{4} (\frac{4}{x})} = 1 \Rightarrow {l o g}_{4} (\frac{4}{x}) = {l o g}_{4} x \Rightarrow \frac{4}{x} = x \Rightarrow x = 2

S o

i t b e c o m e s {l o g}_{4} 2 + {({l o g}_{4} 2)}^{2} + \dots = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots . = 1

Answered by $@y@m last updated on 11/Oct/20

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

1 - \log x = \log x

2 \log x = 1

\log x = \frac{1}{2}

x = 4^{\frac{1}{2}}

x = 2

Commented by ZiYangLee last updated on 12/Oct/20

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