Why do we learn logarithms in a modern era of computers?
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Logarithms are an important concept in mathematics whether or not a computer is available.
For example, if you have an exponential relationship y = a^x and want to express x in terms of y (that is, find the x that makes the exponential relation produce a desired value y) then you’ll need logarithms. This is not some idle task: lots of growth models involve exponential behavior (see radioactive decay, including the formula for half-life, which uses logarithms). Anytime you encounter an important function, its inverse function is probably important too: square roots are inverses of squares and logarithms are inverses of exponentials.
Some more examples: logarithms occur in the description of entropy in thermodynamics and information theory, they are used to describe the decibel scale in acoustics, and they are used to describe the Richter scale for earthquakes. The section on applications of logarithms on the Wikipedia page for logarithms mentions more examples in the real world involving logarithms.
In calculus, logarithms provide the missing formula for an antiderivative of the function x^r when r = -1: the antiderivative for all other values or r is much simpler, but that simpler formula makes no direct sense if r = -1. So if logarithms had not been created before calculus was developed (which they were), they’d have to have been created to make calculus a more complete subject.
There are a bazillion ways logarithms appear in math studied on its own (pure math), but that’s certainly not the kind of answer you are seeking, so I won’t mention those.
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