Are Logarithms Useless? Why Need them Given Exponents?

LOL Why do we need calculators? Aren't they just a crutch to do calculations?

Calculators and computers were prohibited in my schooling and early in my professional career. I can still use a slide rule.

 

Try differentiating ##f(x) = x^x##

 

When you do cover it, you might find the humble logarithm quite useful!

 

You can just view the logarithm as a way to turn multiplicative things into additive things, e.g. Schrodinger used this idea to derive his Schrodinger equation in describing the Hydrogen atom, so it would allow you to work just adding 3 + 6 instead of multiplying 10^3*10^6.

You can also view it's inverse, the exponential e, as a way to turn additive things into multiplicative things.

 

If you work with dB, you will see how logarithms help in the calculations and plotting graphs. Theoretically you can do everything in linear scale of course, it's just not nearly as practical.

It's necessary for doing stuff such as solving ##3 = 10^x## for x, solving certain equations, expressing something in function of something else etc. which is not what I would classify as "useless", but you are just being introduced to the concept and playing around with what it means, in which case it's normal that you don't see its use yet.

 

Once you decide that exponents are needed and inverse functions are needed, you pretty much already have logarithms. Calling them log is just giving a name to something that is already there, namely, the inverse of an exponential function. Another way to say this is that you need logs to solve equations like 10^x = 5. The solution is log 5. You could just say there's some number that solves the equation and have a calculator find it, but it's handy to give it a name, since it comes up often enough. In calculus, the natural log is needed to express the area under the curve y = 1/x. Logarithms come up in computer science, too because if you want to know how long it takes to do a binary search on an ordered list, the answer is log_2 n, where n is the length of the list. A related problem would be the problem of finding how many generations of rabbits are need to produce n baby rabbits, where each rabbit in each generation has 2 babies. The answer is log_2 n again.

Originally, I think logarithms were invented to help do calculations.

 

Imagine my struggles to teach logarithmic meter reading to my data takers.

Practical fission power levels vary over thirteen decades, in my plant's application, from source range Counts Per Second - CPS, through intermediate range detector current amperes, and into one and a half decades of power range indication 1% to 150%.

 

Or, "Same reason we have subtraction and division when we already have addition and multiplication."?