Are logarithms a necessary evil in solving complex problems?

[Cbray]

I don't know why but when I'm solving a huge problem and I end up having to use logarithms I find it frustrating since they take most of my time up since I haven't studied them very often, and they're not very interesting in my opinion.

What's your opinion on logarithms? Do they appeal to anyone, or do you just find them ridiculously annoying to come across?

 

[Office_Shredder]

Do you also hate subtraction, division and taking square roots?

 

[Cbray]

Do you also hate subtraction, division and taking square roots?

I love doing everything else apart from logarithms, I think the reason why I hate them is because I haven't learned them well enough..

 

[jtbell]

If you hate logarithms, then you must also exponential functions, because they go hand-in-hand (they're inverses of each other).

 

[MacLaddy]

I used to dislike them as well, but that was before I really understood what they were all about. I knew they performed an operation, and that it was the inverse of an exponential function, but nobody ever explained what exactly was happening. I ended up sitting down with a whiteboard and played with several log's until I got it. It's simply how many times do you have to divide a number by the base until you end up with the number 1.

This is probably obvious to most people, so I shouldn't admit it, but it really was an epiphany when I figured that out.

Get to know them, and you'll realize they make life much easier.

 

[collinsmark]

I advise taking the time to make friends with logarithms. If you treat them well, they can make your life much easier.

Nearly all of human senses are logarithmic based.

If the lighting is dim, and you want to turn the lights up a bit, you might adjust the dimmer to double the intensity. If that's still not bright enough, you might turn it up a notch by doubling it again (thus 4 times the original intensity).

Do you play a musical instrument? A₄ is 440 Hz. What's one active above that? 2 x 440 Hz = 880 Hz. What's the next octave? Is it 3 x 440 Hz? No it's 4 x 440 Hz. Then 8 x, and so on.

The human ear can hear frequencies approximately from 20 Hz to 20,000 Hz. So what is a "mid-range" frequency? If you guessed 10,010 Hz, you would be incorrect. Even though it's right in the middle in the linear range, it sounds like a high-pitched whine. 10,000 Hz is definitely well into the treble range. Mid-range is actually more like 1000 Hz. That's because humans perceive frequencies logarithmically.

Prick yourself with a pin. Okay, that hurts. What's the next level? Two pins? Okay. Then 4 pins after that for the next level. So what's the next level after that? 5 pins? No. There's not that much difference between 4 and 5 pins. The next level is 8 pins. The "ouch"ness can be expressed as a function of log(N), where N is the number of pins.

Engineers use logarithms all the time, every day, by using the decibel. Decibels are a measure of power ratios, in the form of

$$10 \ \log_{10} \left( \frac{P_o}{P_i} \right)$$

You want to halve the power? Simple: just subtract 3 dB. 'Want to increase the power 10-fold? Just add 10 dB. One hundred fold? 10+10 = 20 dB. How about 200 fold? 10+10+3 = 23 dB.

If you have a 3G or 4G cell phone, its transmit power can vary over 73 dB. That means its maximum output power (when you're at the edge of the coverage area) is more than twenty million times that of its minimum transmit power (when you're right next to a cell tower). Yes, a range of over 20,000,000 times. Working in terms of dB makes everything much easier.

The human ear can hear an enormously large range of volumes. 'Stereo too loud? Turn it down 3 dB. Want to turn it down another notch? Hit it again by 3 dB. That's why volume knobs are always logarithmic. If volume knobs worked on a linear scale, you would be in a world of frustration.

 

[QuarkCharmer]

Great post collinsmark.

 

[BobG]

I like logarithms. My slide rule wouldn't work without them.

Plus, as collinsmark pointed out, most of human senses and perception work on a logarithmic scale instead of a linear scale.

 

[Dickfore]

Every aspiring scientist must understand logarithms and be comfortable using them. When a quantity of interest spans several orders of magnitude, one is obliged to use a logarithmic axis to plot it.

If one axis is logarithmic (the y-axis), and the other is still linear (the x-axis), and we have a linear relationship:
$$\log{Y} = a + b \, X$$
this actually represents an exponential function:
$$Y = \exp(a + b \, X) = \exp(a) \, \exp(b \, X)$$
For example, an exponential decay is a downward sloping straight line on a semi-log plot.

If both axes are logarithmic, and there is a straight relationship between them, then:
$$\log{Y} = a + b \, \log{X}$$

$$Y = \exp \left( a + b \, \log{X} \right) = \exp(a) \, \exp(b \, \log{X}) = \exp(a) \, X^{b}$$
This is a power law.

 

[jim hardy]

i grew up with slide rules so logarithms came as natural as tying my shoelaces.
I still remember from high school math class : log(pi) = 0.49715
Slide rule is an analog computer that multiplies by adding together inches in proportion to the logarithms of numbers to be multiplied.

play with this link a few minutes.
http://www.antiquark.com/sliderule/sim/n909es/virtual-n909-es.html
slide the glass over pi on D cale and read log(pi) on L scale.
Observe 3.162 is smack in middle of D scale and is $\sqrt{10}$

Some people collect slide rules, keep your eyes peeled in thrift stores.

If i encountered space aliens i'd try to start the conversation by showing them my old slide rule. Its markings demonstrate visually that we count in base ten and are aware of transcendental functions.

I hope you get to like logarithms.

old jim

 

[Astronuc]

Logrithms have their uses and I rarely enounter them. Most of the time now, I spend with complex computer codes and simulations.

BTW - John Napier developed logarithms in 1614. The were the precursor to slide rules.
http://ualr.edu/lasmoller/napier.html

The slide rule was invented/developed in 1632!
http://www.cs.kent.edu/~rothstei/10051/History.htm
http://history-computer.com/CalculatingTools/logarythms.html

 

[BobG]

Some people collect slide rules, keep your eyes peeled in thrift stores.

I have quite a few. In fact my avatar is cropped from a picture of my little pocket Pickett.

 

[TestTubeGames]

I have to agree with BobG and others above: Slide Rules are the reason I enjoy logs.

Cbray, if you haven't checked one out, I encourage it. (I raided my parents stash of them and now have my own collection). It is a triumph of simple & clever over complex. Sure, I'm a child of the computer age, and calculators are far more useful. But with no transistors, no electronics, just two pieces of plastic that can slide past one another... you can do most (if not all) of what a scientific calculator can.

It blew my mind the first time I saw it. That weird abstract math equation about how logs add suddenly became not only clear but useful.

 

[BobG]

you can do most (if not all) of what a scientific calculator can.

A good duplex slide rule, such as a Post 1460 (the Post Versalog), does more than a standard scientific calculator does. You have to move up to the graphing calculators before you have a calculator that does more than a slide rule.

You can solve quadratic equations and solve equations using complex numbers on a slide rule - something most scientific calculators can't do easily (actually, you could use the same method on your scientific calculator that you use on a slide rule for complex numbers, but it would just be harder). On the other hand, your graphing calculators usually have a polynomial solver that allows you to solve even higher order polynomials.

And try solving this on a calculator (it can be done at least on a graphing calculator, but I'm not so sure with your standard scientific calculator).

$$e^x=x^4$$

 

[Astronuc]

I used Keuffel & Esser (K+E) slide rules, but I may have had a Pickett also.
The one I used in university had the scales in this image: http://www.sphere.bc.ca/test/slide6/221-k&e-68-1210.jpg

 

[turbo]

I used Keuffel & Esser (K+E) slide rules, but I may have had a Pickett also.
The one I used in university had the scales in this image: http://www.sphere.bc.ca/test/slide6/221-k&e-68-1210.jpg

I had a very similar K&E (LOTS of scales!). We were forbidden to use calculators in engineering school because even the cheapest 4-function calculators cost about half of a semester's tuition. Still, once you learned your way around a slide rule, you could out-perform anybody who was using a calculator.

Years later, working as a process chemist in a pulp mill, I had my little book of log tables, and other little "bibles" of engineering standards. Sometimes, the old ways are not discredited, but are superseded by ways that are more convenient or easier to use by the uninitiated. I'll bet BobG could tear apart lots of engineering problems with his slide rules.

 

[skeptic2]

My chemistry prof had a cylindrical one that was also a pencil holder.

 

[Dickfore]

A good duplex slide rule, such as a Post 1460 (the Post Versalog), does more than a standard scientific calculator does. You have to move up to the graphing calculators before you have a calculator that does more than a slide rule.

You can solve quadratic equations and solve equations using complex numbers on a slide rule - something most scientific calculators can't do easily (actually, you could use the same method on your scientific calculator that you use on a slide rule for complex numbers, but it would just be harder). On the other hand, your graphing calculators usually have a polynomial solver that allows you to solve even higher order polynomials.

And try solving this on a calculator (it can be done at least on a graphing calculator, but I'm not so sure with your standard scientific calculator).

$$e^x=x^4$$

The solutions of this equation are expressible in terms of the Lambert W-function ($W(z) \, \exp(W(z)) = z$):
$$x = \log(x^4) = 4 \, \log(x)$$
$$\frac{\log(x)}{x} = \frac{1}{4}$$
$$\frac{1}{x} \, \log \left( \frac{1}{x} \right) = -\frac{1}{4}$$
$$y \equiv \log \left(\frac{1}{x}\right) \Rightarrow x = \exp(-y)$$
$$y = W(-\frac{1}{4}) \Rightarrow x = \exp(-W(-1/4))$$
Since the Lambert W-function is a multiple valued function with infinitely many branches, the above equation has infinitely many solutions. Here's a list of the first 11:
From Wolfram Alpha

 

[BobG]

Perhaps WolframAlpha is computing the complex solutions in addition to the real solutions (for some reason, it seems to be taking an infinite amount of time for the solutions to appear).

If one were to graph the two functions, there wouldn't be an infinite number of intersections.

(Plus, WolframAlpha isn't a standard scientific calculator - it's a CASS program.)

 

[Dickfore]

If one were to graph the two functions, there wouldn't be an infinite number of intersections.

That's not true, since the exponential function $\exp(z)$ is periodic in the complex plane with a period $2 \pi i$. Within one period (a band with sides parallel to the real axis and width $2\pi$), the modulus of this function takes all positivie values (as the real part goes from $-\infty$ to $\infty$). The function $z^4$ has an unbounded modulus, and it winds the phase four times, so it will have many (I am thinking four) intersections in each band. That's infinitely many intersections.

 

[Dickfore]

Here is a semilog plot of {Re[x],-Im[x]} (I had to use the minus sign since all the imaginary values turn out to be negative) of the first 1001 complex roots to that equation:

 

[jim hardy]

I used K&E also, the Decilon. Calculators weren't out yet.

Two engineers at GE published an article in design magazine ca 1966 describing a calculator they'd built based on rate-multipliers. Title was "Electronic Digital Slide Rule " and i have searched for it to no avail.
It did basic four function (no floating point) and they described how to do transcendental functions with rate multipliers. Theirs was made with the IC's of the day, just a few gates per chip, and occupied an entire fishiing tackle box. Much of that was lantern battery and inverter for the Nixie tube displays.

I have been fascinated by rate multipliers since. The old RCA datasheets had best explanation of using them for transcendentals.

Anyhow the article said that within a decade they'd be small enough for a shirt pocket.

I didnt believe them.

I didnt buy Ebay's IPO either.

 

[Danger]

I had to look up "logarithm".

Now that I know what they are, I don't much like them. I do, however, encounter them. One of my hobbies is designing and building firearms suppressors, and decibels are on a logarithmic scale. I've never used math, though; I just go by what it sounds like.