How do I solve this equation involving a logarithm?xlog⁡(x)=100

Question

umkhululiueyj · Accepted Answer

You will need the services of the Lambert function

W (x)

to solve this equation. Briefly, the Lambert function is the inverse of the function

x e^{x}

: if

x = y e^{y}

, then

y = W (x)

.
To turn your equation into a form where the Lambert function's appearance becomes transparent, let's first turn everything into natural logarithms:

x \ln x = 100 \ln 10

and then we make the left side a "little" complicated:

(\ln x) e^{\ln x} = 100 \ln 10

We now recognize the Lambert form, and thus perform the inversion:

\ln x = W (100 \ln 10)

from which

x = e^{W (100 \ln 10)} \approx 56.961248432 \dots

oanhtih6 · Accepted Answer

If xlogb(y)=z then taking anti-logarithms you get yx=bz.So in this case with y=x and b=10 you get xx=10100.You will not find it easy to solve this explicitly for x; try reading about the Lambert W function or use numerical methods to get something just over 56.96.

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