Solve for X

Since this problem concerns exponents, we think of taking the logarithm on both sides. $ln x^{x^{x...}} = ln 10$ becomes $x^{x^{x...}} ln x = ln 10$ . Now, the pre-logarithmic term, $x^{x^{x...}}$ , is equal to 10. Thus, $10 ln x = ln 10$ which gives us $x = e^{\frac{ln 10}{10}}$ , which is actually $x = 10^{\frac{1}{10}} = 1.259...$

$simply\quad substitute\quad the\quad power\quad of\quad x\quad to\quad 10\quad since\quad { x }^{ { x }^{ { x }^{ \infty } } }=10\\ your\quad equation\quad right\quad now\quad is\quad { x }^{ 10 }=10,\quad \\ by\quad this\quad equation,\quad you\quad can\quad now\quad solve\quad the\quad value\quad of\quad x\\ x=\sqrt [ 10 ]{ 10 } \\ x=1.259$

I know most of you already know this, but for the sake of the others, here's my brief solution ${ x }^{ 10 }=10\\ x=\sqrt [ 10 ]{ 10 } \simeq 1.259$