A student made an error while proving the following claim that he believed to be true. Claim: The number $\pi$ equals the number $e$.\ The student's proof: Let $f ( x )$ and $g ( x )$ be functions for $x > 0$ defined as $\mathrm{f} ( \mathrm{x} ) = \ln ( \pi \mathrm{x} )$ and $\mathrm{g} ( \mathrm{x} ) = \ln ( \mathrm{ex} )$.\ I. For every $x > 0$, the derivatives of functions $f ( x )$ and $g ( x )$ are equal to each other.\ II. Therefore, for every $x > 0$, functions $f ( x )$ and $g ( x )$ are equal to each other.\ III. Since $\ln ( x )$ is one-to-one and $f ( x ) = g ( x )$, we conclude that for every $x > 0$, $\pi x = ex$.\ IV. If two functions are equal for every $x > 0$, then their values at $x = 1$ are the same.\ V. Since the values of the functions $\pi \mathrm{x}$ and $ex$ at $x = 1$ are the same, we conclude that $\pi = \mathrm{e}$.\ In which of the numbered steps did this student make an error?\ A) I\ B) II\ C) III\ D) IV\ E) V
A student made an error while proving the following claim that he believed to be true.
Claim: The number $\pi$ equals the number $e$.\
The student's proof: Let $f ( x )$ and $g ( x )$ be functions for $x > 0$ defined as $\mathrm{f} ( \mathrm{x} ) = \ln ( \pi \mathrm{x} )$ and $\mathrm{g} ( \mathrm{x} ) = \ln ( \mathrm{ex} )$.\
I. For every $x > 0$, the derivatives of functions $f ( x )$ and $g ( x )$ are equal to each other.\
II. Therefore, for every $x > 0$, functions $f ( x )$ and $g ( x )$ are equal to each other.\
III. Since $\ln ( x )$ is one-to-one and $f ( x ) = g ( x )$, we conclude that for every $x > 0$, $\pi x = ex$.\
IV. If two functions are equal for every $x > 0$, then their values at $x = 1$ are the same.\
V. Since the values of the functions $\pi \mathrm{x}$ and $ex$ at $x = 1$ are the same, we conclude that $\pi = \mathrm{e}$.\
In which of the numbered steps did this student make an error?\
A) I\
B) II\
C) III\
D) IV\
E) V