- #1
DocZaius
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I saw a YouTube video presenting what seemed like a clever solution to ##x^{x^{x^{.^{.}}}} = 2## (which is to say: an infinite tower of exponents of x = 2). He said to consider just the exponents and ignore the base and realize that those exponents themselves become a restatement of the whole left side of the equation. Equate them to 2 and you get ##x^2=2## and thus ##x= \sqrt{2}##.
I thought that was clever and tried to see if this could work for other operations like multiplication (##x\cdot x\cdot x \cdot\ ...=2##), division (##\frac{x}{\frac{x}{\frac{x}{...}}}=2##), addition (##x+x+x+...=2##) and subtraction(##x-x-x-...=2##), but realized it didn't. Any idea what's special about the exponent operation that allows for this clever "substitution of infinite terms" type of solution?
I thought that was clever and tried to see if this could work for other operations like multiplication (##x\cdot x\cdot x \cdot\ ...=2##), division (##\frac{x}{\frac{x}{\frac{x}{...}}}=2##), addition (##x+x+x+...=2##) and subtraction(##x-x-x-...=2##), but realized it didn't. Any idea what's special about the exponent operation that allows for this clever "substitution of infinite terms" type of solution?