It's interesting how "useful day to day" can change so much in context. Logarithms are the basis for how slide rules work so in the time before personal computers when logarithms were more useful then than today.

And while most people don't actively use logarithms in their day to day life, they are incredibly important because human perception is generally logarithmic, not linear. The decibel scale is logarithmic because of this. There's even some evidence to suggest that logarithmic thinking might even be more natural. https://news.mit.edu/2012/thinking-logarithmically-1005

Imaginary number is a bad name for them... and intentionally so. It was meant to be derogatory. Mathematics has a history of people not liking new developments because people think of math as a very logical and objective thing and those developments can fly in the face of what they believe.

There's an apocryphal story that someone in the Cult of Pythagoras proved that the square root of two is irrational and they were so outraged by the idea that a number could be irrational that they took him to sea in a boat and returned without him. They believed that all numbers could be expressed as a ratio of two integers and an irrational number by definition is one that can't be expressed as a ratio.

The Ancient Greeks also didn't believe in the idea of zero or negative numbers and both were very controversial in the Western world for many centuries afterwards. In math today, the standard form for polynomials to put all the coefficients on one side and set it to zero because it's really useful. For example, Ax^2 + Bx + C = 0 for the quadratic polynomial where B and C are allowed to be zero and A/B/C are all allowed to be negative. But up until I think the 1500s, Western mathematicians didn't have a standard form but a family of forms specifically to avoid zeroes and negative numbers. Ax^2 + Bx = C, Ax^2 = Bx + C, Ax^2 = C, Ax^2 = Bx, etc.

Imaginary numbers first saw real use in the cubic equation because the people who found it realized that in some cases it involved taking the square root of a negative number, which people believed to be nonsensical. However, the cubic equation worked because the imaginary numbers cancelled each other out. Thus, they were called imaginary because people didn't think they were "real" and were only a mathematical trick that happened to work out and not something that's meaningful.

To get a feel for what it was probably like when irrational numbers, zero, negative numbers, and imaginary numbers were first introduced, look at the comments whenever 1+2+3+4+5+6+... = -1/12 comes up.