Interpreting meaning of log answer

M

markgoldman

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I understand that the meaning of log[sub:rez5nkaq]10[/sub:rez5nkaq] 1000 = 3 is that (10) (10) (10) = 1000. But I don't quite understand the meaning of log[sub:rez5nkaq]10[/sub:rez5nkaq] 1100 = 3.0413. Can you explain and show what that means--that is, what is being multiplied by itself 3.0413 times to get 1100? I can't quite understand what it means to multiply something by itself 3.0413 times. And can you show me to do the calculation that generates the log of 1100?

Thanks very much.

Mark
 
\(\displaystyle log_{10}1000 \ = \ 3 \ \implies \ 10^3 \ = \ 1000.\)

\(\displaystyle log_{10}1100 \ = \ 3.0413 \ \implies \ 10^{3.0413} \ = \ 1100.\)

\(\displaystyle log_{10}10,000 \ = \ 4 \ \implies \ 10^4 \ = \ 10,000.\)

\(\displaystyle Common \ sense, \ if \ 10^3 \ = \ 1000 \ and \ 10^4 \ = \ 10,000, \ then \ 10^{3.0413} \ must \ lie \ in \ between.\)

\(\displaystyle log_{10}1100 \ = \ x, \solve \ for \ x.\)

\(\displaystyle Easy \ way, TI-89. \ 10^x \ = \ 1100, \ x \ \dot= \ 3.04139268516\)

\(\displaystyle Hard \ way, \ book \ of \ common \ logs \ to \ four \ places. \ Note, \ since \ the \ advent \ of \ calculators,\)

\(\displaystyle "the \ hard \ way" \ has \ become \ obsolete, \ thank \ God.\)

\(\displaystyle log_{10}1100 \ = \ x, \ solve \ for \ x.\)

\(\displaystyle Whipping \ out \ my \ "trusty \ book \ of \ common \ logs", \ I \ note \ that \ the \ number \ 1100 \ correspond\)

\(\displaystyle to \ the \ mantissa \ 0414. \ Since \ its \ characteristic \ is \ 3, \ we \ have \ 3.0414, \ no \ interpolation,\)

\(\displaystyle I \ got \ lucky.\)
 
markgoldman said:
what is being multiplied by itself 3.0413 times to get 1100? I've never liked this terminology.

I can't quite understand what it means to multiply something by itself 3.0413 times. Don't try to understand it too deeply. You might go nuts.
Click to expand...

Hi Mark:

I would not say that 10^3 is 10 multiplied by itself three times. To me, that would be 10^4.

Instead, I say that 10^3 is the product of three factors of 10.

In other words, when an exponent is a Whole number, it tells me how many factors of the base are multiplied together.

10^3 means three factors of 10

10^4 means four factors of 10

10^3.0414 means three factors of 10 and a "mystery" factor.

< mmm4444bot envisions pka rolling his eyes, right about now >

Well, the value of this mystery factor is no cliffhanger; it has to be 11/10 .

10 * 10 * 10 * 11/10 = 1100

So, in my mind, the mystery comes into play because I also know that all of the factors must be powers of 10.

10 * 10 * 10 * 10^x = 1100

x must be 0.0414 because 10^0.0414 = 11/10

10 * 10 * 10 * 10^0.0414 = 10^3 * 10^0.0414 = 10^(3 + 0.0414) = 1100

At this point, I would suppose that your next question is something like, "Okay. Then tell me what it means to say 10^0.0414 because I don't understand how a power of 10 can have less than one factor of 10."

To which I would respond, "When we grow beyond the Whole numbers, we're not in grade school, anymore. Concepts become more deep. Try thinking about the continuity of exponential graphs and the denseness of the set of Real numbers, to feel at ease with "fractional" powers smoothly increasing in value over a range of infinite exponents. Beyond that, don't try to understand it too deeply."

Although, weird concepts are fun to consider. I'm glad to see that you're thinking; we don't always get much of that around here.

Do you think that 16^0.5 = 4 is weird ? I don't, anymore. When I see 16^0.5 now, it seems natural. 8-)

Cheers ~ Mark
 
Thank your for the response. Rather than just blurt something back, if it's ok I'd like the luxury of thinking about your observations. Your comment about did I think that 16^0.5 is weird is right on point, and got me thinking. "Obviously" I don't think it's weird, so I'm wondering why a fractional log bothers me but a fractional exponent doesn't, especially when I realize logs and exponents are essentially rearrangements of one another. I've not previously used this forum, so I don't know the etiquette here. May I get back to you with a follow-up in a day or so, or do things here just move on to the next topic?

In any event thanks very much for taking the time to get back to me.
 
markgoldman said:


logs and exponents are essentially rearrangements of one another Sort of.

log = exponent

I would say that logs and exponents are two different notations for the same concept. We write logs more often when we either don't know the exponent or we want to express an irrational exponent exactly.

To what exponent do we need to raise 10, in order to get 40? Off the top of one's head, who knows? But, we can write down that exponent anyway:

log(40)



May I get back ... with a follow-up in a day or so You betcha. I don't read every new post here, but I look at them all.

I only operate at two speeds, anyways. Slow, and slower.
Click to expand...
 
\(\displaystyle For \ what \ it's \ worth \ Dept.\)

\(\displaystyle The \ history \ of \ logarithms \ is \ a \ story \ in \ itself.\)

\(\displaystyle I \ won't \ get \ into \ it \ but \ for \ anyone \ interested, \ just \ google.\)
 


Hi Mark. Have you gone nuts, yet ? :wink:

You can still respond if and when you like; there's no rush.

I just want to add some more comments about the meaning of 10^0.0414, for your consideration.

(0.0414 is not the exact exponent. It is rounded.)

This approximate exponent can be written as 414/10000.

So, 10^(414/10000) can also be thought of as the 10,000th root of 10^414.

\(\displaystyle 10^{3.0414} \;=\; 10^3 \cdot\sqrt[10000]{10^{414}}\)

Cheers,

~ Mark

 
Noah's boat finally comes to rest as the flood waters recede, and he lowers the gangway and send the animals out calling to them, "Go forth and multiply".

Most of the animals leave, but two snakes are left behind. Noah looks at them, and commands "Go forth and multiply!"

The snakes look at him but do not move. He tries again, "Go forth and multiply!" The snakes do not move.

Noah gets angry and in his most commanding voice shouts, "Go forth and multiply!"

The snakes look up at him and say, "We can't, we're adders".

Noah thinks for a while, then grabs his saw and hammer and runs off into the forest, where he cuts down a tree. He saws and hammers and builds a small table. He carefully picks up the snakes and puts them on the table.

"Go forth and multiply!" he commands.

The snakes look at each other, and then at Noah. "We can't, we're adders".

"Yes", Noah replies, "but, even adders can multiply on a log table".
 
OK Subhotosh, you asked for it!

THAT'S WHY YOU'RE HERE
======================
All that existed was the creator, God. And he was perfect and sinless;
that's easy enough to understand: there was nobody around to play or
sin with. Well, we can safely assume that God got bored being all over
the place with nothing to do. He must have said to himself something
like "oh my God...this is getting real boooooring...gotta find myself
something to do..."

So God moseyed over to the holy lab, mixed up a few cosmic elements,
said magic words like "abra-cadabra", made a holy "pouff!" with his
lips and "bingo!"...or should I say "bigBANG!"...out came the universe.
This kept him out of mischief for a while...but watching big heavenly
basketballs rotating around themselves gets boring after a while.
"Jumpin Jehosapha...this is getting to be real boring" God must have
said to himself. So he let go a couple of spiritual yawns...and then
returned to the holy lab.

We can safely assume that God sat himself down to do some serious
thinking, something along the line of "gee whizz, it might be fun to
re-produce myself in some new form and move to one of them planets...
ya, why not...it'll give me someone to watch over and care for". So he
pulled out a few of his holy genes, dropped them in a test-tube, then
experimented patiently...until out came a satisfactory product which
he named "Adam".

God then crazy-glued stars to the sky, picked a planet, fixed it up
for human habitation, named it "Earth", then deposited Adam on it...
somewhere close to suburban Jerusalem. After watching Adam do his thing
for a while, God noticed that loneliness was setting in..."oh me, oh my,
I'm so lonely" wailed poor Adam.

"My God" God thought to himself, "I'd better fix him up with a partner".
So he reached down, pulled a rib out of Adam. "Jesus Murphy, that hurts!"
yelled poor Adam. On his way back to the holy lab with Adam's rib, God
was scratching his head..."did he swear?" he wondered, "and who the ****
is Jesus Murphy?"

God experimented with Adam's rib, shook it around in a test-tube, and
"POP!"...maybe I should say "MOM!"...out jumped Adam's partner. However,
something went haywire in the lab: what came out was a weirdly shaped
thing talking at over a hundred miles an hour. Well, God reluctantly
decided to keep it, called it "woman"...short for "Watch Out MAN"...
and named it "Eve".

God then figured out that it would be easier to re-produce himself by
using his two new human creations. Yep, he made up his mind to do this,
muttering to himself "good Lord, I got better things to do than running
back to this stupid lab every time I want to re-produce myself".

So he equipped Eve with a gadget designed to excite Adam, then dropped
her beside wide-eyed Adam. "Wanna fool around, big boy?" said Eve,
wetting her lips. Well, the very first chase took place...which, by the
way, is still going on today, better known as "chasing woman".

Well, Eve eventually tripped...believed to be on purpose...Adam imitated
Tarzan, but then was not sure what to do. God saw this, jumped in, issued
instructions on the act and positions (to which quite a few have been
added since!) and equipped them both with a relief mechanism which he
named "the orgasm".

And that's why you're all here today.
 
mmm444bot: Thank you very much. Your comment that 10^0.0414 can be thought of as the 10,000th root of 10^414 is the insight/frame of reference I needed. The moment I read your comment I thought "...well of course dummy, that is what it means" And of course that's the explanation for the decimal component in the log that got me started on this question in the first place. It's funny--I could perform the operation/get the "right" answer, but for some reason, when I was reading the book that led to the question, I couldn't satisfactorily explain to myself what the answer meant--as you could tell from my question. It was very frustrating having the answer but not understanding conceptually what the answer meant.

So the insight you gave me is quite a relief. Please accept my appreciation for the time and trouble you took to help me out. It was particularly thoughtful of you to return to the question and offer an alternative way of looking at it. That was really unexpected.

Thanks, and best regards,

Mark
 
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