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When I write graph for fun I found that there is some value a that make $y=a^x$ and $y=\log_a (x)$ don't touch each other and there is some value a that make $y=a^x$ and $y=\log_a (x)$ intersect each other too (2 intersect point). Because of this there is a value a that make they touch each other.

https://sv1.picz.in.th/images/2019/05/03/wKD9ry.jpg

So, I try to find that value by writing graph. Then, I get a = 1.4444... or 13/9. But I have no idea why a = 13/9.

Please help me or guide how to prove that.

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    $\begingroup$ It is not $\frac {13}9$ but $e^{\tfrac 1e}\approx 1.44466786100977$ instead. $\endgroup$ – Mohammad Zuhair Khan May 2 at 17:17
  • $\begingroup$ Thank you. How to calculate a ? $\endgroup$ – uesdto signin May 2 at 18:08
  • $\begingroup$ A useful paper for this and closely related issues: Nikola Koceić Bilan and Ivan Jelić, On intersections of the exponential and logarithmic curves, Annales Mathematicae et Informaticae 43 (2014), 159-170. $\endgroup$ – Dave L. Renfro May 2 at 20:12
  • $\begingroup$ If $a^x=\log_a x \implies x=a^{a^x}$. Solutions of $x = a^x$ are also solutions to $x = a^{a^x}$ so you can achieve some solutions with the Lambert-W function. $\endgroup$ – Mohammad Zuhair Khan May 2 at 20:25

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