Simplify the expression $\log_2({x_1}^{y_1} + {x_2}^{y_2})$ for large values of $x,y$

Asked Apr 20 '21 at 11:12

Active Apr 20 '21 at 11:12

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Following this thread: Simplify the expression $\log(2^x+2^y)$ for large values of $x,y$

Is there a way to simplify (or get a good approximation) for the $\log_2$ of a sum of powers with different bases?

Formally, I'm trying to simplify the following expression: $\log_2({x_1}^{y_1} + {x_2}^{y_2})$, where $x_1 \neq x_2$ and $y_1 \neq y_2$ (or in general for $x_1,...x_n$ and $y_1, ... y_n$), for cases where $y_1, y_2$ are very large and the power can't be computed.

asked Apr 20 '21 at 11:12

Ofir Gordon

With $,x,y\gg1,$ use $,x+y\approx\max(x,y).$ – Somos Apr 20 '21 at 12:08
@Somos It's possible but this is a too loose approximation if for instance $x \gg y$ – Ofir Gordon Apr 22 '21 at 07:19
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Note $\log_2({x_1}^{y_1} + {x_2}^{y_2}) = \log_2\Big(2^{y_1\log_2(x_1)}+ 2^{y_2\log_2(x_2)}\Big)$ and use the methods in the linked thread. – Somos Apr 22 '21 at 11:37

Simplify the expression $\log_2({x_1}^{y_1} + {x_2}^{y_2})$ for large values of $x,y$

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