Logs

Links

Eddie Woo has a series of videos (I'm not necessarily impressed with this guy):

• logs, introduction, part 1; link here.
• logs, introduction, part 1: link here.

 

• understanding logs, part 1: link here.
• understanding logs, part 2: link here.

 

• laws of logs, part 1: link here.
• laws of logs, part 2: link here.
• laws of logs, part 3: link here.

 

• changing bases: link here.

 

• graphs of logs, part 1: link here.
• graphs of logs, part 2: link here.
• graphs of logs, part 3: link here.

 Logarithms

How do you say this:  

• log10 100 = 2: this is read as 'log to the base 10 of 100 is 2'. 

I find that unhelpful.

How about this: 

• "The logarithm of 100 is 2; or, which is never said, "2 is the logarithm of 100."
• what everyone says: "the log of 100 is 2."

The "base" is generally unsaid and assumed to be "10" unless otherwise specified, but if the base is necessary, then this is how you say it:

"The log to the base (of) ten of 100 is 2.

Another way to think about this this or look at this function (equation):  

• log10 100 = x: solving for x, how many times must 10 be multiplied by itself to obtain the value of 100?
  

Ten must be multiplied by itself twice (10 x 10) to get to 100.

I doubt Napier ever said anything like that. He probably simply looked at "100" and then knew immediately and for numbers like 53, instead of 100, he simply looked at his beautiful table, or abbreviated it to say "log100=."

Another way to look at this -Truemper, 2020, p. 89:

log b (y) = x if b^x = y

this is how you say that, the log of y is x if y equals b^x

 So, log 10 (1000 = 3 because 1000 = 10^3.

Seems convoluted, hard to say, hard to remember, simply remember / memorize the formula. 

Why do we have different bases, such as 10, 2, e?

Base 10: probably historical. That's the number Napier (and others) first used when they "discovered" logarithms. And, so, in general, "10" is the default base. Interestingly enough, is base ten even that relevant any more in the computer age? See below?

Base 2: in computer science the base "2" is often used because computer language is binary, yes/no, off/on. Apparently, in other disciplines, like biology, base 2 is also often common.

Base e: interestingly enough and perhaps for unknown reasons, the magic number e often shows up when working complicated formulas in physics. One can convert logs from base e to base 10 but log base e comes up so often in physics that it's easier to simply keep base e to begin with.

Note: e is so commonly used and comes up so often "naturally" in the discipline of physics and e is also known as the "natural logarithm." It is interesting that "base 10" is not called the "common logarithm" or the Briggs logarithm. LOL.

Why do we need to know how to change the "base" when calculating logarithms. Because scientific calculators have room for only so many buttons,  manufacturers agreed to limit logarithm bases to two: base 10 and base e (the natural log number; Euler's number).

It seems, in addition to base 10 and base e, there's only one other base that commonly occurs, base 2, which begs the question: wasn't there enough room for just one more base? LOL. 

Note: scientific computers have only so much room for clickable keys. Makers have generally agreed to have to "log" keys -- one for the "natural logarithm," e; and, one for the "common logarithm," base 10. I suppose if they marketed a "scientific calculator for biology students," maybe it would have a key for base 2. LOL.  

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Rules: Only Seven

Logarithm rules: link here.

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Actually: Only Eight Rules
The last rule.

Changing bases:

log b (x) = y

log b (x) = log 10 (x) / log 10 (b)

So, using "log x" button on any scientific computer,

• find log x -- write it down, numerator
• find log b -- write it down, denominator
• divide numerator / denominator and one gets a number that is the answer to log b.

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YouTube very, very basic: link here.

YouTube basic: link here.

YouTube natural logarithm, very interesting: link here. This one is fascinating.

The history of the derivation of Euler's number: link here.

Why Euler's number is so important:

The exponential function is equal to its own derivative: link here. Great video. Incredible. Is this proof that there is a God?

 

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History - Narrative

Why did Napier and others explore the math that eventually let to logarithms?

• multiply 3,594,874 x 2,385,238; or,
• add 6.334 + 6.124 take that answer, go to the published tables, and get the result?

Where (in what discipline) did Napier, 1550 - 1617 (William Shakespeare, 1564 -1616), find the need for a simpler / better way to multiply / divide large numbers: astronomy; naval navigation. He did this by hand. How many numbers did he calculate and how long did it take?

Do logarithms work as well for really, really small numbers, just as it works for very large numbers? Explain.

Who discovered the "natural base," e? What was this mathematician working on when he came across this number that eventually came to be called the "natural log." This mathematician did not call this number e and initially (if ever) realized it was related to logarithms. 

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