How can we find the inverse of this function? $$y=\exp(ax)+\exp(bx),$$ where $a$ and $b$ are constants
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In general, there is no closed form for it. For example, if $a=5$, $b=1$, write $t=e^x$, then you have $t^5+t-y=0$, and it's well-known you can't solve for $t$ in terms of elementary functions.