I don't know how to solve this. I have tried base change but it's not working. Please provide a solution.
2026-03-25 22:30:53.1774477853
How I can solve $\log_{\frac{1}{(2+|x|)}}(5+x^2)=\log_{(3+x^2)}(15+\sqrt x)$
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Note that right side is only defined for $x\ge 0$, so we may restrict our analysis there. Now, on the right side, $15+\sqrt x \gt 1$ so right hand side is always positive.
On the other hand, left side, we have for $x> 0$, $$\log_{\tfrac{1}{1+2|x|}}(5+x^2) = -\log_{1+2x}(5+x^2)$$ and since $1+2x < 5+x^2$, so $\log_{1+2x}(5+x^2)$ is always positive and left hand side is always negative. There are no solutions.