I'm just a high school guy so this question is hopeless for me. I don't know advanced mathematics. I'm trying to get $x$ in terms of a. When I try to cancel the $\log$ by substitution, exponential functions appear. Is there some way to at least approxumate $x$ in terms of $a$ by using infinite series or anything else?
Solution of the equation $\frac{\log x}{x}=1-\frac{1}{a}$
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sequences-and-series
functions
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2There isn't a closed form for x in terms of elementary functions. Your solution will involve the Lambert W function. Solving the simplified equation log(x)/x = a results in x = -W(-a)/a. – 2017-01-31
1 Answers
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Let $y=1/x$. Then taking the antilogarithm,
$$ye^y=e^{1/a-1}$$ which is solved by the Lambert function
$$\frac1x=y=W\left(e^{1/a-1}\right).$$
You won't find a simpler expression.
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0What's Lambert function? What is it's formula or infinie series? – 2017-01-31
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1@Dove: Wikipedia is your friend. – 2017-01-31