Why do we learn logarithms in a modern era of computers?
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Are you asking why we use them outside of computation, which we no longer need them for?
There are lots of applications and a big one is optimization.
Basically, say we have some expression we wish to optimize – find the maximum or minimum of. There are well known ways to do this via calculus… but in many cases they become easier if we instead optimize the logarithm of that expression.
This is because logarithms turn products into addition which are easier to work with. And since log is an increasing function, optimizing the log of an expression is the same as optimizing the expression itself.
Lots of the current work in deep learning/AI for example is built, essentially, on optimizing expressions involving logarithms. Programming this requires understanding the underlying optimization problem and that includes being able to manipulate logarithms.
But that’s just one examples. There are many others in theoretical or applied math.
Are you trying to get someone to answer your homework?
The logarithm is a common and useful mathematical function that shows up in a ton of different places. Logarithms aren’t just there for slide rules. They show up as a common integral in calculus (integral du/u = ln(u) + C) and thus in the solution to a very common type of differential equation. They show up when solving any equation involving a variable in an exponent, as with things like half-lives, decay constants, or continuously compounding interest. A full list of everywhere they appear would be quite long.
ELI5: What are logarithms?
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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