Jamiya Weber

2022-06-26

Show combination of affine functions and logs has at most one zero
For $x>0$, let
$f\left(x\right)=\left(x+2\right)\mathsf{l}\mathsf{o}\mathsf{g}\left(x\right)-\left(x+1\right)\mathsf{l}\mathsf{o}\mathsf{g}\left(x+1\right)$
Can anybody show that the equation $f\left(x\right)=0\left(x>0\right)$ has at most one solution.

pheniankang

Expert

We compute
${f}^{\prime }\left(x\right)=1+2/x+\mathrm{log}x-1-\mathrm{log}\left(x+1\right)=2/x-\mathrm{log}\left(1+1/x\right).$
We want to show that this is positive. Putting $y=1/x$, we just need to show $2y>\mathrm{log}\left(1+y\right)$ for all positive y.
But these two functions are equal when $y=0$ and the result is then clear by the concavity of the logarithm.

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