Mathematics

Problem

A differentiable function

y

satisfies

\dfrac{dy}{dx}=\dfrac{1}{x}

and passes through the point

(3,7)

. A student writes the antiderivative

y=\ln|x|+C

and then replaces

\ln|x|

with

\ln x

to obtain

y=\ln x+C

. Which of the following statements best justifies this replacement? a) Because the constant

C

can absorb any sign change, the absolute value bars are unnecessary. b) Because the initial condition requires

x=3>0

, the particular solution is confined to

x>0

, where

|x|=x

. c) Because the natural logarithm is defined for all real numbers, the absolute value bars may always be removed. d) Because

\ln|x|

and

\ln x

have the same derivative

\dfrac{1}{x}

for

x>0

, replacing

|x|

with

x

is always valid.