Question 185823
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \log{x^{\frac{1}{2}}}=sqrt{\log{x}}]


First use *[tex \Large \log_b{x^n}=n\log_b{x}] to write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{1}{2} \log{x}=sqrt{\log{x}}]


Multiply by 2:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \log{x}=2 sqrt{\log{x}}]


Square both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  (\log{x})^2=4\log{x}]


Divide by *[tex \Large \log{x}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \log{x}=4]


Use *[tex \Large \log_b{x}=y \ \ \Leftrightarrow\ \ b^y = x] to write:



*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x = 10^4]


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*[tex \LARGE \ \ \ \ \ \ \ \ \ \  N(a) = 5000 = 2000 + 500 \log{a}]



*[tex \LARGE \ \ \ \ \ \ \ \ \ \  500 \log{a} = 3000]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \log{a} = 6]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  a = 10^6]

John
*[tex \LARGE e^{i\pi} + 1 = 0]
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