Answer
$f(x)=\ln|x|+1.$
Work Step by Step
The tangent line at $(x, f(x))$ has slope $f'(x)$, which is given:
$f'(x)=\displaystyle \frac{1}{x}$
So, $f(x)$ is an antiderivative:
$f(x)=\displaystyle \int\frac{1}{x}dx=\ln|x|+C.$
The indefinite integral gives us a collection of functions.
To find the exact function, we find C.
We are given that $f(1)=1$, so
$1=\ln 1+C$
$1=0+C$
$C=1$
So,
$f(x)=\ln|x|+1.$