The main calculation in practical astronomy is solving a spherical triangle, from three elements one wants to find the rest, using formulas of spherical trigonometry.
This calculation is always performed when
for example one transfers between the coordinate systems.
Positions of celestial bodies are usually described in ecliptic coordinates, while observations are made in equatorial coordinates with stationary telescopes,
and in practical astronomy (navigation) horizontal coordinates must be used. A typical formula used in navigation looks like this:
$$\sin h=\sin\phi\sin\delta+\cos\phi\cos\delta\cos t,$$
where $\phi$ is latitude, $\delta$ is the declination $t$ is the hour angle, and $h$ the altitude of a celestial body. To find the ship position with the most common method, using a chronometer, one has to use this formula at least twice (for two stars, for example). Without a chronometer, a more complicated calculation is required.
Under ordinary circumstances,
for example in navigation one computes each element of a triangle in degrees, minutes and fractions of a minute. (In the past they used seconds, but in 20th century decimal fractions of a minute were more commonly used in navigation.) Since 90 degrees=5400 minutes=324000 seconds,
one needs 5-6 digits for the needs of practical astronomy/navigation.
Naked eye and practical astronomy do not allow measuring of angles with higher accuracy. In scientific astronomy, the accuracy is higher, since with the invention of telescope they measured angles to fractions of a second, which means already 7-8 digits.
Now try to multiply two random 5 digit numbers by hand, and you will see why logarithms were such a big discovery. And a typical astronomical calculation involves many such multiplications.
By the way the first log tables (of Napier) tabulated not $\log x$ but $\log\sin x$, that is they were specially designed for trigonometric calculations in astronomy/geodesy.