Log third

Algebra Level 3

For some constant $a> 1$ , the two functions $f(x) = a^x$ and $g(x) = \log_a x$ intersect exactly once.

What is $a?$

e^{1 }

e^{ -1 }

e^{e }

e^{ -e }

e^{ \frac{1}{e} }

1 solution

Prince Loomba
May 1, 2016

$We\quad know\quad that\quad these\quad functions\quad are\quad inverse\quad of\quad each\quad other\\ and\quad inverse\quad functions\quad are\quad mirror\quad images\quad about\quad y=x.\\ So\quad wherever\quad they\quad intersect,\quad their\quad slopes\quad will\quad be\quad equal\quad to\quad 1.\\ a^{ x }=\frac { lnx }{ lna } ...(1)(by\quad base\quad changing\quad theorem).\\ a^{ x }lna=1....(2),\\ \frac { 1 }{ xlna } =1...(3).\\ Using\quad these\quad 3\quad equations,\\ from\quad 3,xlna=1,lna=\frac { 1 }{ x } ,e^{ lna }=e^{ \frac { 1 }{ x } },\boxed { a=e^{ \frac { 1 }{ x } } } .\\ Using\quad 1,(a^{ x }=\frac { lnx }{ lna } );\\ from\quad 2,a^{ x }lna=1,lna=\frac { 1 }{ a^{ x } } .\\ Substituting\quad this\quad value\quad in\quad the\quad equation\quad 1,\quad lnx=1,\quad x=e.\\ Putting\quad x=e\quad in\quad the\quad obtained\quad equation,we\quad get\quad a={ e }^{ \frac { 1 }{ e } }\\$

Log third

1 solution

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