Explaining Logarithm

What is logarithm?

The logarithm of a number is -more or less- how many digits it has.

So the logarithm of 83472388347238 is about 77 (actually it's 6.921....6.921....). The logarithm of 99 is about 11. So it is a convenient way to measure the scale of a number.

Now this was a very rough idea. Let's make it more precise. If you take any number and multiply it by 1010,it gains an additional digit, so according to what I said above, the logarithm increases by one. So what actually happens is, the logarithm of a number is how often I have to multiply 1010 to get that number. So log(100)=2\log_{(100)}=2 because 10×10=10010 \times 10 = 100, so I multiplied two 10s10s together. Similarly, log(10000)=4\log_{(10000)}=4 because 10×10×10×10=1000010 \times 10 \times 10 \times 10=10000, and so on. Multiplying four 10s10s together can also be written as 10410^4. And now we can generalise the idea:

log(x)=y\log_{(x)}= y is another way of saying that 10y=x10^y = x.

The final question you may have is: what's so special about the number 1010? Why does it count how many 10s10s I have to multiply together? Well actually, logarithm is used with other base numbers as well: base 22 and base ee are also very common. But for each choice, you get logarithm function that measures the scale of the number you put in.

What are logarithms used for? Are decibels a good example of the usefulness of a logarithm? Are logarithms calculus?

Logarithms can be used to talk about things that can be both tiny and gigantic, such as in

- earthquake magnitudes,

- noise levels in decibels, and

- acidity (pH).

A big earthquake can be millions of times bigger than a tiny one. If you tried to make a bar graph where the bars has sizes 10,100,10, 100, and 1000000010 000 000, it would look stupid. The bars of size 1010 and 100100 would be too small to see, and you won't be able to tell that one of them is ten times bigger than the other. If you instead take the logarithm of each number, you get 1,2,1, 2, and 7.7. That makes a bar graph you can understand.

Keep that in mind when you hear about earthquake magnitudes. A 7.07.0 earthquake is ten times bigger than a 6.06.0 earthquake, which is ten times bigger than a 5.05.0 earthquake. Taking logarithms lets us put an earthquake caused by a stick of dynamite (1.2)(1.2) on the same scale as the 20112011 earthquake in Japan (9.0)(9.0).

Logarithms can also be used to measure how long it will take something to grow exponentially or decay exponentially,such as

- money growing with a fixed interest rate,

- bacteria growing in a petri dish,

- radioactive decay, for example in Radiocarbon dating, and

- the sound made by a bell.

If you have bacteria that divide every 3030 minutes and are currently taking up 0.10.1% of the petri dish, you can use logarithms to estimate how long it will take them to fill up the entire dish. The same goes for $5000\$5000 in an account with a 22% interest rate. If you leave the interest in the account, logarithms will tell you when you'll have $6000\$6000.

Logarithms can also be used in calculations by turning multiplication into addition. If I gave you the option between multiplying twenty numbers together by hand or adding twenty numbers together by hand, you'd pick the second option. If you need to multiply twenty numbers, you can instead take the base 1010 logarithm of each number, add the results, and then raise 1010 to that power. Finding the logarithm might seem hard, but, in the past, people could just look it up in a logarithm table or use a slide rule. Finding the answer using logarithms was way faster.

The problem of multiplying lots of numbers was the original reason logarithms were developed. This method is now obsolete thanks to computers, which are pretty fast at multiplying. In the meantime, though, we've discovered tons of uses of logarithms, most of which I haven't even listed here.

What is the intuition behind the logarithm?

The logarithm counts the number of groupings.

Suppose a bakery puts 1212 cookies in a package, and places 1212 of these packages in a larger box for transport:

Then a box can be seen as cookies which have been grouped twice: one box contains 122=14412^2 = 144 cookies. Inversely, when ordering 144144 cookies, and knowing that this bakery works with base 12,12, the logarithm will return the number of groupings: log12144=2\log_{12}{144} = 2 groupings. Suppose the transport company also works in base twelve; 1212 boxes are wrapped in plastic, 1212 plastic units are stacked onto a wooden pallet, 1212 pallets are transported in a van, which makes 123=172812^3=1728 boxes per ride:

Now the total number of cookies in one transport, is obtained by multiplying the number of cookies per box with the number of boxes per ride: 144×1728=248832144 \times 1728 = 248832 cookies.
However, if we look at this on a grouping scale (logarithmic scale) then we must use addition instead: the total number of groupings is 2+3=52 + 3 = 5. Stated otherwise: log12144+log121728=log12248832\log_{12}{144} + \log_{12}{1728} = \log_{12}{248832}. Now suppose 2073620736 packages of cookies were stolen. Then we know that this involved log1220736=4\log_{12}{20736} = 4 groupings. So counting upwards from these packages...

...we know that this monster learned how to drive:

Note by Cera Mess
3 years, 4 months ago

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How do I setup a logarithm for calculating the increase from interest rate or that petri dish thing you mentioned? I've always wondered that.

The Strategy Gamer - 3 years, 4 months ago

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For the petri dish example, at

t=0t=0 the petri dish is 0.10.1% full,

t=30mint=30min the petri dish is 0.10.1% ×2\times 2 full,

t=60mint=60min the petri dish is 0.10.1% ×22\times 2^2 full,

t=90mint=90min the petri dish is 0.10.1% ×23\times 2^3 full,

t=120mint=120min the petri dish is 0.10.1% ×24\times 2^4 full.

You can see that a general formula for how full the petri dish at time tt is 0.10.1% ×2(t(30min))\times 2^{\left(\frac{ t }{ (30min)}\right )}. For example, for t=120min,t(30min)=4,t=120min, \frac{t }{(30min) }= 4, and indeed the petri dish is 0.10.1% ×24\times 2^4 full.

Assuming this growth rate continues until the petri dish is full, to find when that happens, we want to find the time t so that 0.10.1% ×2(t(30min))=100\times 2^{\left(\frac{t} {(30min)}\right)} = 100%. We begin by dividing both sides by 0.10.1% to get 2(t(30min))=1000.2^{\left( \frac{t }{ (30min) }\right)} = 1000. Then we take the base two logarithm of both sides. t(30min)=log2(1000)\frac{t}{ (30min)} = log_2(1000) Then we multiply both sides by 30min30min to get t=(30min)×log2(1000)t = (30min) \times log_2(1000), or about 55 hours.

Cera Mess - 3 years, 4 months ago

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Thanks! It's easier than any logarithmic equation exercise they teach us, and yet no one says anything about how useful this is in real life. Too bad high school algebra is all like that.

The Strategy Gamer - 3 years, 4 months ago

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Using logarithms is still a useful way to multiple numbers. While for small numbers, it's probably faster to just multiply - as the computer probably has a dedicated circuit for it - for larger numbers (numbers with a lot more bits than the word size), using logarithms to change it into an addition problem is probably going to be faster. This is especially true the less we care about the precise accuracy, and our numbers are too large to represent as floating point numbers (which has good hardware support).

Joms Joseph Leelin - 3 years, 4 months ago

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" This is especially true the less we care about the precise accuracy, and our numbers are too large to represent as floating point numbers (which has good hardware support)"

I didn't get it?

Jeff Ishee - 3 years, 4 months ago

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Floating point doubles top out around 10^300, and even quads top out around 10^5000. Doing calculations on larger numbers requires other methods.

Rafael Melo - 3 years, 4 months ago

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