I sympathise with you. It certainly seems that those people who do best at mathematics in school are just those who are best at abstract manipulation of symbols.
The example you give is an interesting one. Logarithms are quite hard to motivate[+]. Once they could be introduced as analogous to a slide rule as something to speed up multiplication (indeed that was their actual original motivation). But that isn’t convincing these days. Perhaps a motivation could be found in science (something about gradients and log-log plots). The problem is that the real reason you learn about logs is because they are related to exponential and exponential are important. Consider the differential equation:
dy/dx = k y
This kind of equation comes up a lot in modelling things in science and other parts of the real world (eg compound interest). We can solve it:
k dx/dy = 1/y
k x = log(y) + C
log(y) = k x - C
y = exp(k x - C)
y = A exp(k x) (where A = exp(-C) is chosen to fit initial conditions)
But this motivation isn’t much use when one doesn’t have differential equations or calculus. When I was in school I think differential equations where only in A-level further maths (age 16-18). I’m not sure whether we had logarithms at GCSE (14-16) but I think they were in A level maths which included calculus up to integration by parts and integrals/derivatives of various trig functions.
[+] By motivate I mean “put into context of some practical application”.
In the “real world” such mathematical things aren’t necessarily well motivated either. It may be that one has to calculate certain weird things because that is what clients/customers/taxes/regulations/circumstances dictate.
In applied mathematics things aren’t so well motivated. Either “we see these equations in physics sometimes” or for fluids you mumble something about boats or planes. In pure mathematics the motivation basically always boils down to “we need this crazy thing to solve a certain problem to tell if two objects are the same/different/somehow interesting because we are interested in this sort of object and in particular how to classify the different kinds of it.