I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources:

• Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute convergence, we have that $$\frac{d}{ds}= \sum_{n=1}^{\infty}\frac{a(n)\log n}{n^s}$$
• The function $\Gamma(s)$ is ubiquitous in analytic number theory (one need not look beyond the functional equation of $\zeta(s)$), and $$\frac{\Gamma'}{\Gamma}(z) = \log z -\frac{1}{2z}+ O(z^{-2}).$$
• Mellin inversion, which allows us to express partial sums of arithmetic functions in terms of contour integration, is (essentially) a logarithmic change of variables away from Fourier inversion.

This does not account for everything (these are largely motivated by multiplicative number theory and connections with $L$-functions), but it accounts for a lot, including the asymptotic

$$\sum_{p\leq x}\frac{1}{p} = \log\log x + B + O((\log x)^{-1})$$.

This single asymptotic accounts for a lot of the presence of $\log x$ and its higher compositions in sieve theory. Combine these with ideas like partial summation, and the logs accumulate.

I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources:

• Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute convergence, we have that $$\frac{d}{ds}\sum_{n=1}^{\infty}\frac{a(n)}{n^s}= -\sum_{n=1}^{\infty}\frac{a(n)\log n}{n^s}$$
• The function $\Gamma(s)$ is ubiquitous in analytic number theory (one need not look beyond the functional equation of $\zeta(s)$), and $$\frac{\Gamma'}{\Gamma}(z) = \log z -\frac{1}{2z}+ O(z^{-2}).$$
• Mellin inversion, which allows us to express partial sums of arithmetic functions in terms of contour integration, is (essentially) a logarithmic change of variables away from Fourier inversion.

This does not account for everything (these are largely motivated by multiplicative number theory and connections with $L$-functions), but it accounts for a lot, including the asymptotic

$$\sum_{p\leq x}\frac{1}{p} = \log\log x + B + O((\log x)^{-1}).$$

This single asymptotic accounts for a lot of the presence of $\log x$ and its higher compositions in sieve theory. Combine these with ideas like partial summation, and the logs accumulate.

I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources:

• Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute convergence, we have that $$\frac{d}{ds}= \sum_{n=1}^{\infty}\frac{a(n)\log n}{n^s}$$
• The function $\Gamma(s)$ is ubiquitous in analytic number theory (one need not look beyond the functional equation of $\zeta(s)$), and $$\frac{\Gamma'}{\Gamma}(z) = \log z -\frac{1}{2z}+ O(z^{-2}).$$
• Mellin inversion, which allows us to express partial sums of arithmetic functions in terms of contour integration, is (essentially) a logarithmic change of variables away from Fourier inversion.

This does not account for everything (these are largely motivated by multiplicative number theory and connections with $L$-functions), but it accounts for a lot, including the asymptotic

$$\sum_{p\leq x}\frac{1}{p} = \log\log x + B + O((\log x)^{-1})$$.

This single asymptotic accounts for a lot of the presence of $\log x$ and its higher compositions in sieve theory. Combine this with ideas like partial summation, and the logs accumulate.

I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources:

• Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute convergence, we have that $$\frac{d}{ds}= \sum_{n=1}^{\infty}\frac{a(n)\log n}{n^s}$$
• The function $\Gamma(s)$ is ubiquitous in analytic number theory (one need not look beyond the functional equation of $\zeta(s)$), and $$\frac{\Gamma'}{\Gamma}(z) = \log z -\frac{1}{2z}+ O(z^{-2}).$$
• Mellin inversion, which allows us to express partial sums of arithmetic functions in terms of contour integration, is (essentially) a logarithmic change of variables away from Fourier inversion.

This does not account for everything (these are largely motivated by multiplicative number theory and connections with $L$-functions), but it accounts for a lot, including the asymptotic

$$\sum_{p\leq x}\frac{1}{p} = \log\log x + B + O((\log x)^{-1})$$.

This single asymptotic accounts for a lot of the presence of $\log x$ and its higher compositions in sieve theory. Combine these with ideas like partial summation, and the logs accumulate.