There are several ideas here, some mentioned by in the other answers.:
One: When Gauss was a boy (By by the dates found on his notes he was approximately 16) he noticed that the primes appear with density $\frac{1}{\log x}$ around $x$. Then, instead of counting primes and looking at the function $\pi (x)$, lets weight by the natural density and look at $\sum_{p\leq x} \log p$. Since we are weighting by what we think is the density, we expect it to be asymptotic to be $x$.
Two: Differentiation of Dirichlet series. If $$A(s)=\sum_{n=1}^\infty a_n n^{-s}, \text{ then } A^'(s)=-\sum_{n=1}^\infty a_n \log (\log n) n^{-s}$$ The $\log$ term appears naturally in the differentiation of Dirichlet series. Taking the convolution of $\log n$ with the Mobius function (that is multiplying by $\frac{1}{\zeta(s)}$) then gives the $\Lambda(n)$ mentioned above. The $\mu$ function is really the special thing here, not the logarithm.
Expanding on this, there are other weightings besides $\log p$ which arise naturally from taking derivatives. Instead we can look at $\zeta^{''}(s)=\sum_{n=1}^\infty (\log n)^2 n^{-s}$, and then multiply by $\frac{1}{\zeta(s)}$ as before. This leads us to examine the sum $$\sum_{n\leq x} (\mu*\log^2 )(n)$$ (The $*$ is Dirichlet convolution) By looking at the above sum, Selberg was able to prove his famous identity which featured in was at the center of the first elementary proof of the prime number theorem:
$$\sum_{p\leq x} \log^2 p +\sum_{pq\leq x}(\log p)(\log q) =2x\log x +O(x).$$
Three: The primes are intimately connected to the zeros of $\zeta(s)$, and countour contour integrals of $\frac{1}{\zeta(s)}$. (Notice it was feature featured everywhere here so far) We can actually prove that
$$\sum_{p^k\leq x} \log p= x -\sum_{\rho:\ \zeta(\rho)=0} \frac{x^{\rho}}{\rho} +\frac{\zeta^'(0)}{\zeta(0)}.$$ Notice that the above is an equality, which is remarkable since the left hand side is a step function. (Somehow, at prime powers all of the zeros of zeta conspire and make the function jump.)
