Jamiya Weber

Answered

2022-06-26

Show combination of affine functions and logs has at most one zero

For $x>0$, let

$f(x)=(x+2)\mathsf{l}\mathsf{o}\mathsf{g}(x)-(x+1)\mathsf{l}\mathsf{o}\mathsf{g}(x+1)$

Can anybody show that the equation $f(x)=0(x>0)$ has at most one solution.

For $x>0$, let

$f(x)=(x+2)\mathsf{l}\mathsf{o}\mathsf{g}(x)-(x+1)\mathsf{l}\mathsf{o}\mathsf{g}(x+1)$

Can anybody show that the equation $f(x)=0(x>0)$ has at most one solution.

Answer & Explanation

pheniankang

Expert

2022-06-27Added 22 answers

We compute

${f}^{\prime}(x)=1+2/x+\mathrm{log}x-1-\mathrm{log}(x+1)=2/x-\mathrm{log}(1+1/x).$

We want to show that this is positive. Putting $y=1/x$, we just need to show $2y>\mathrm{log}(1+y)$ 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.

${f}^{\prime}(x)=1+2/x+\mathrm{log}x-1-\mathrm{log}(x+1)=2/x-\mathrm{log}(1+1/x).$

We want to show that this is positive. Putting $y=1/x$, we just need to show $2y>\mathrm{log}(1+y)$ 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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