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Some Quadratics Equals Linear (Posted on 2024-11-30)

Use Lambert's W function to solve this equation:

2^x = 10*x.

No Solution Yet Submitted by K Sengupta

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Solution

Comment 1 of 1

Just hammer on the equation for a bit

2^x = 10x

10x * 2^(-x) = 1

(-10\ln2)*(-ln2*x) * e^(-ln2*x) = 1

(-ln2*x) * e^(-ln2*x) = -(ln2)/10

Now apply W

-ln2*x = W(-(ln2)/10)

x = -W(-(ln2)/10)/ln2

-1/e < -(ln2)/10 < 0 is true, so then the principal branch W_0 and branch W_-1 will both be real numbers.

Then Wolfram Alpha evaluates x as solutions x_0 ~= 0.107755 and x_-1 ~= 5.87701

Posted by Brian Smith on 2024-11-30 14:00:07

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