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anonymous

  • 5 years ago

What is the sum of the real solutions to x^(logx) = (x^3)/100 ?

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  1. anonymous
    • 5 years ago
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    first take the log of both sides to get the variable out of the exponent. \[log(x^{log(x)})=log(\frac{x^3}{100})\] then use the properties of logs so put everything on the ground floor. \[log(x)log(x)=log(x^3)-log(100)=3log(x)-2\] hope it is clear than log(100)=2 since 10^2=100 \[(log(x))^2=3log(x)-2\] now you have a quadratic equation (in log(x) so set = 0 and solve \[(log(x))^2-3log(x)+2=0\] this one factors. \[(log(x)-2)(log(x)-1)=0\] two solutions are \[log(x)=2\] and \[log(x)=1\] now find x \[x=10^1\] \[x=10^2=100\] total is 110

  2. anonymous
    • 5 years ago
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    Thanks, I remember learning this in the beginning of the year now!

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