Home -> Solved problems -> Can you solve it? x^2 = 2^x
Solution
This visual solution to the equation \(x^2 = 2^x\) explains the real solutions step by step using logarithms and the Lambert W function. Each step is illustrated clearly, so you can follow the reasoning and understand how the equation is solved intuitively. By studying this approach, you’ll gain a deeper understanding of the relationship between exponential and polynomial functions.
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\[
\begin{aligned}
x^{2} &= 2^{x} \\
x &\neq 0 \\
\ln(x^{2}) &= \ln(2^{x}) \\
2 \ln|x| &= x \ln 2 \\
x^{-1} \ln|x| &= \frac{1}{2} \ln 2 \\
\text{case (1): } x &> 0 \\
x^{-1} \ln x &= \frac{1}{2} \ln 2 \\
\text{Using Lambert W function:} \\
W(-\ln x \, e^{-\ln x}) &= W(-\ln \sqrt{2}) \\
-\ln x &= W(-\ln \sqrt{2}) \\
x &= e^{-W(-\ln \sqrt{2})}
\end{aligned}
\]
\[
\begin{aligned}
\text{case (2): } x &< 0 \\
(-x)^{-1} \ln(-x) &= -\frac{1}{2} \ln 2 \\
W(-\ln(-x) \, e^{-\ln(-x)}) &= W(\ln \sqrt{2}) \\
-\ln(-x) &= W(\ln \sqrt{2}) \\
x &= -e^{-W(\ln \sqrt{2})}
\end{aligned}
\]
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