# Are Logarithms Useless? Why Need them Given Exponents?

Just curious, since we're discussing them this week in my class. Why do we need logarithms? Aren't they just a concoction to express exponents?

Doug Huffman
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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.

PeroK
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Just curious, since we're discussing them this week in my class. Why do we need logarithms? Aren't they just a concoction to express exponents?
Try differentiating ##f(x) = x^x##

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

Don't know what you mean. Haven't covered that yet.

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

Mark44
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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.
Yeah, me, too. I still own about five of them.

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

Don't know what you mean. Haven't covered that yet.
Well, then you have probably covered exponential functions such as y = ex, y = 10x, and the like. Log functions, in an appropriate base, are the inverses of the exponential functions. For example,

##y = 10^x## is equivalent to x = log10(y) = log(y) and ##y = e^x## is equivalent to x = loge(y) = ln(y).

By "equivalent to" I mean that the graphs of y = 10x and x = log(y) are identical.

The concept of logs is also used for plotting data when either the "x" or "y" values (or both) cover a range of values of several orders of magnitude. Such plots are done on semi-log paper or log-log paper.

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.

Nugatory
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You can just view the logarithm as a way to turn multiplicative things into additive things

Entropy, for example. It is really convenient to be able to add entropies.

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.

Doug Huffman
Gold Member
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%.

Chestermiller
Mentor
It isn't just an arithmetic tool. The solutions to an enormous number of actual real world problems in heat transfer, fluid mechanics, chemical reaction kinetics, solid mechanics, economics, finance, and thermodynamics are expressed directly in terms of natural logarithms.

Chet

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

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