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## Is there a way to calculate logarithm with a base of 10 on paper?

• Posted by a hidden member.
So I can do it with a calculator.

To recap: log(base b)(x) = y, thus b^y = x

So log(base 10)28 = 1.447 (using a calculator), as 10 to the 1.447 power equals 28.

Is there a way to answer this example on just a sheet of paper and without a calculator? It's really hard to calculate powers for me as I'm not sure how they increments (sort of exponentially?)

um......GEEKS UNITE!
• Posted by a hidden member.
Ok I better not see anyone complaining when I aimlessly pick on plebeians and flame the crap out of them as I usually do ...because I tried to have an intellectual conversation and you all failed ...miserably ...to entertain me! Such tyros. Anyway, I'm done being a pedant. On to masturbation.

• Posted by a hidden member.
It's bloody hard to work it out on paper.

Just go the old school route if you really want to do it.

http://en.wikipedia.org/wiki/Slide_rule
• Posted by a hidden member.
• Posted by a hidden member.
Our Calculus teacher never taught us how to derive log manually (same with the trigonometric functions - sine, cosine, tangent, cotangent, secant, cosecant). It was a magic thing you required a calculator to get. -_- Although our trigo teacher did teach us how to derive the circular functions manually so that excused him for that part... but log remains a mystery only computers get.

• kew1

Posts: 1621
Nor did mine, but I'm older & used log tables.
• Posted by a hidden member.
JakeBenson saidOk I better not see anyone complaining when I aimlessly pick on plebeians and flame the crap out of them as I usually do ...because I tried to have an intellectual conversation and you all failed ...miserably ...to entertain me! Such tyros. Anyway, I'm done being a pedant. On to masturbation.

Isn't masturbation done by the "common people" and a person who makes an excessive or inappropriate display of learning just thinks about masturbation?

Which would explain why the pedant is so tense and needs a really good anal penetration?
• Posted by a hidden member.
There's a reason calculators were created. Embrace technology!
• Posted by a hidden member.
JakeBenson saidSo I can do it with a calculator.

To recap: log(base b)(x) = y, thus b^y = x

So log(base 10)28 = 1.447 (using a calculator), as 10 to the 1.447 power equals 28.

Is there a way to answer this example on just a sheet of paper and without a calculator? It's really hard to calculate powers for me as I'm not sure how they increments (sort of exponentially?)

um......GEEKS UNITE!

I used to know how to do that shit but my college weed days and drinking have erased that shit.
• Posted by a hidden member.
jakebensonIt's really hard to calculate powers for me as I'm not sure how they increments (sort of exponentially?)

Concerning the exponent:

You can change the 1.447 to a fraction (1447/1000) then,

Whenever you see a fractional exponent, remember that the top number is the power, and the lower number is the root (if you're converting back to the radical format).

10^1447/1000 can be written as 1000rt 10^1447

or look at this thread about base 10 logs in physics...hope it helps

• groundcombat

Posts: 945
Nope sorry. The only way I remember being able to do logs without a calculator is either to just be familiar enough with the base to have them memorized or guess and check.
Both of which are annoying.
• FlashDrive

Posts: 53
JakeBenson saidSo I can do it with a calculator.

To recap: log(base b)(x) = y, thus b^y = x

So log(base 10)28 = 1.447 (using a calculator), as 10 to the 1.447 power equals 28.

Is there a way to answer this example on just a sheet of paper and without a calculator? It's really hard to calculate powers for me as I'm not sure how they increments (sort of exponentially?)

um......GEEKS UNITE!

I don't know what kind of paper calculation you meant, but if you have some constants handy and know the expansion of log(e base), you pretty much can evaluate any logarithm numbers by hand; such as following:

Constants you need: log(base 10)(2), log(base 10)(3), log(base10)(e),etc
Expansion:
log(base e)( 1 + x ) = x - (1/2)*(x)^2 + (1/3)*(x)^3 - (1/4)*(x)^4 + (1/5)*(x)^5 - ...
where e = 2.71828 ... [commonly log(base e)x = lnx ] and x^2<1
also recall: log(base 10)(X) = log(base 10)(e) * log(base e)(X)

Ex:
log(base 10)28 = log(base 10)(2x10x1.4) = log(base 10)2 + log(base 10)10 + log(base 10)(1.4)
[comment: the key is to reduce the argument of log to a "(1+x)" fasion, where x<1. In here, it is 1.4=1+0.4];
now
log(base 10)(2) = 0.30103
log(base 10)(10) =1
log(base 10)(1.4) => this needs to use expansion above
only to the 5th term acuuracy here
=> log(base 10)(e) * ( (0.4) - (1/2)*(0.4)^2 + (1/3)*(0.4)^3 - (1/4)*(0.4)^4 + (1/5)*(0.4)^5 )= (0.43429)*(0.33695)=0.146334
where log(base 10)(e) = 0.43429

finally
0.30103 + 1 + 0.146334 = 1.447364
(if you go to 6th order, number is like 1.44707)
the calculator gives 1.447158, which means there is a chance to converge to this number if higher oder terms >7 are used.

What a scenic route to get log(base 10)28 !! it sounds silly.
Maybe the main takeaway is that the lograthrim can always be broken down to smaller numbers or known numbers to be manageable.
• Posted by a hidden member.
FlashDrive saidI don't know what kind of paper calculation you meant, but if you have some constants handy and know the expansion of log(e base), you pretty much can evaluate any logarithm numbers by hand; such as following:

Constants you need: log(base 10)(2), log(base 10)(3), log(base10)(e),etc
Expansion:
log(base e)( 1 + x ) = x - (1/2)*(x)^2 + (1/3)*(x)^3 - (1/4)*(x)^4 + (1/5)*(x)^5 - ...
where e = 2.71828 ... [commonly log(base e)x = lnx ] and x^2<1
also recall: log(base 10)(X) = log(base 10)(e) * log(base e)(X)

Ex:
log(base 10)28 = log(base 10)(2x10x1.4) = log(base 10)2 + log(base 10)10 + log(base 10)(1.4)
[comment: the key is to reduce the argument of log to a "(1+x)" fasion, where x<1. In here, it is 1.4=1+0.4];
now
log(base 10)(2) = 0.30103
log(base 10)(10) =1
log(base 10)(1.4) => this needs to use expansion above
only to the 5th term acuuracy here
=> log(base 10)(e) * ( (0.4) - (1/2)*(0.4)^2 + (1/3)*(0.4)^3 - (1/4)*(0.4)^4 + (1/5)*(0.4)^5 )= (0.43429)*(0.33695)=0.146334
where log(base 10)(e) = 0.43429

finally
0.30103 + 1 + 0.146334 = 1.447364
(if you go to 6th order, number is like 1.44707)
the calculator gives 1.447158, which means there is a chance to converge to this number if higher oder terms >7 are used.

What a scenic route to get log(base 10)28 !! it sounds silly.
Maybe the main takeaway is that the lograthrim can always be broken down to smaller numbers or known numbers to be manageable.

Rawr! That's one of the sexiest things posted on here in years.

People who know me know I'm not kidding.
• Erikkkk

Posts: 285
iguanaSF said
FlashDrive saidI don't know what kind of paper calculation you meant, but if you have some constants handy and know the expansion of log(e base), you pretty much can evaluate any logarithm numbers by hand; such as following:

Constants you need: log(base 10)(2), log(base 10)(3), log(base10)(e),etc
Expansion:
log(base e)( 1 + x ) = x - (1/2)*(x)^2 + (1/3)*(x)^3 - (1/4)*(x)^4 + (1/5)*(x)^5 - ...
where e = 2.71828 ... [commonly log(base e)x = lnx ] and x^2<1
also recall: log(base 10)(X) = log(base 10)(e) * log(base e)(X)

Ex:
log(base 10)28 = log(base 10)(2x10x1.4) = log(base 10)2 + log(base 10)10 + log(base 10)(1.4)
[comment: the key is to reduce the argument of log to a "(1+x)" fasion, where x<1. In here, it is 1.4=1+0.4];
now
log(base 10)(2) = 0.30103
log(base 10)(10) =1
log(base 10)(1.4) => this needs to use expansion above
only to the 5th term acuuracy here
=> log(base 10)(e) * ( (0.4) - (1/2)*(0.4)^2 + (1/3)*(0.4)^3 - (1/4)*(0.4)^4 + (1/5)*(0.4)^5 )= (0.43429)*(0.33695)=0.146334
where log(base 10)(e) = 0.43429

finally
0.30103 + 1 + 0.146334 = 1.447364
(if you go to 6th order, number is like 1.44707)
the calculator gives 1.447158, which means there is a chance to converge to this number if higher oder terms >7 are used.

What a scenic route to get log(base 10)28 !! it sounds silly.
Maybe the main takeaway is that the lograthrim can always be broken down to smaller numbers or known numbers to be manageable.

Rawr! That's one of the sexiest things posted on here in years.

People who know me know I'm not kidding.

That was hot.
• Posted by a hidden member.
Jakebenson, you have my admiration for your inteligence to even ask such a question. Just for the fun of it, what the hell is a logarithm ? I was in business management for 30 years and never had a need for it. Can this knowledge of and use of a logarithm be of value in my everyday life? How is this knowledge put to use? What line of work uses this knowledge? Mathematicians? Scientists? Architects? I'm not trying to be a smart ass either, I'm trying to learn something.
• Posted by a hidden member.
I haven't done calculus in awhile but.. I think a Taylor Series could pull it off. I hated them though, but pretty sure you don't need a calculator...

realifedad - Logarithms are mostly just for hard core calculus and such... as you can probably tell. Something it can be used for is extremely large numbers, like distance between solar systems etc. Another common example would be pH levels.

pH = log ( 1 / # of hydrogen ions as expressed in moles )

10 ^ pH = 1 / # of hydrogen ions

so a pH of 4 (acidic)...

10 ^ 4 = 1 / # of hydrogen ions

# of hydrogen ions = 0.0001 moles (per unit volume)

This means that for every single digit different the pH, its a factor of 10 in the # of hydrogen ions. So most people dont care how many thousands of ions... they just want a scale to determine if something will burn their eyes out or not.

Disclaimer - I am not a chemist... just a dirt engineer.
• Posted by a hidden member.
Hey thanks for taking the time to tell me some about this logarithm subject, now I at least know something about how its used, but I'll readily admit its way beyond me. LOL !!!
• Posted by a hidden member.
Logarithms are everywhere, man! The logarithm function is the inverse of the exponential function: If y grows exponentially with x, then x grows logarithmically with y. The Richter magnitude scale is logarithmic, as is the pH scale, as CrownRoyal points out. The logarithm also appears in nature, in the form of the logarithmic spiral.

The most familiar mathematical application of the logarithm is the slide rule, owing to the fact that the log of a product is the sum of the logs of the factors. This means that a multiplication task can be converted into an addition task. It is also the basis of the following punny joke:

Noah opens up the ark and lets all the animals out, telling them to "Go forth and multiply." He's closing the great doors of the ark when he notices that there are two snakes sitting in a dark corner.

So he says to them, "Didn't you hear me? You can go now. Go forth and multiply."

"We can't," said the snakes, "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".

In college quantitative courses you'll see the logarithm frequently used as a device to simplify a problem involving multiplication. However, the logarithm also appears unexpectedly in some truly profound and beautiful results in pure mathematics, such as the prime number theorem, which asserts that if you randomly select a number in the vicinity of n, the probability is roughly 1/log n that you got yourself a prime. Also the law of the iterated logarithm is a remarkable result about fluctuations in a random walk. It sez: almost surely, the random walk will be bounded by, and will approach arbitrarily closely, a curve shaped like sqrt(2n log log n) -- infinitely often.
• Posted by a hidden member.
zotamorf62 saidLogarithms are everywhere, man! The logarithm function is the inverse of the exponential function: If y grows exponentially with x, then x grows logarithmically with y. The Richter magnitude scale is logarithmic, as is the pH scale, as CrownRoyal points out. The logarithm also appears in nature, in the form of the logarithmic spiral.
...

In college quantitative courses you'll see the logarithm frequently used as a device to simplify a problem involving multiplication. However, the logarithm also appears unexpectedly in some truly profound and beautiful results in pure mathematics, such as the prime number theorem, which asserts that if you randomly select a number in the vicinity of n, the probability is roughly 1/log n that you got yourself a prime. Also the law of the iterated logarithm is a remarkable result about fluctuations in a random walk. It sez: almost surely, the random walk will be bounded by, and will approach arbitrarily closely, a curve shaped like sqrt(2n log log n) -- infinitely often.

Stop it. I'm getting a permanent hard-on.
• Posted by a hidden member.
NERD ALERT - shit I feel like my Dad and brother are engineering something right before our eyes............ah!
• Posted by a hidden member.
FlashDrive said
Maybe the main takeaway is that the lograthrim can always be broken down to smaller numbers or known numbers to be manageable.

FlashDrive has it right. However, the power series for log(1+x) converges relatively slowly. The name of the game is to reduce your problem to calculating log(1+x) for x as close to zero as possible; you'll get a good approximation in fewer terms. For example, you could write

log10(28 ) = log10(28/25) + log10(25) = log10(1.12) + 2 log10(5)

Dispatch the first term using the power series approximation. Look up the second term, or use log10(5) = 1 - log10(2).

(Geeks unite!)
• jrs1

Posts: 4388
FlashDrive said
JakeBenson said... GEEKS UNITE!

... Constants you need: log(base 10)(2), log(base 10)(3), log(base10)(e),etc
Expansion:
log(base e)( 1 + x ) = x - (1/2)*(x)^2 + (1/3)*(x)^3 - (1/4)*(x)^4 + (1/5)*(x)^5 - ...
where e = 2.71828 ... [commonly log(base e)x = lnx ] and x^2<1
also recall: log(base 10)(X) = log(base 10)(e) * log(base e)(X)

... Maybe the main takeaway is that the lograthrim can always be broken down to smaller numbers or known numbers to be manageable.

I absolutely enjoyed that. especially the aspect and use of the series ...

it makes me reminiscent of the Taylor expansions, where:

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... + x^n/n!, for all x.

... bringing me to the hypergeometric series:

... leading to the aspect of the double integral:

which makes me nostalgic for the simple yet always appealing system reliability component to mathematical probability:

ex: Dr. Kishor S. Trivedi: http://people.ee.duke.edu/~kst/

ex 3: Bayesian application(s) of mathematical probability and reliability: http://www.ncbi.nlm.nih.gov/pubmed/3385460

ex 4: Serial & Parallel Systems: http://www.ecs.umass.edu/ece/koren/FaultTolerantSystems/simulator/NonSerPar/nsnpframe.html

• Posted by a hidden member.
JakeBenson saidSo I can do it with a calculator.

To recap: log(base b)(x) = y, thus b^y = x

So log(base 10)28 = 1.447 (using a calculator), as 10 to the 1.447 power equals 28.

Is there a way to answer this example on just a sheet of paper and without a calculator? It's really hard to calculate powers for me as I'm not sure how they increments (sort of exponentially?)

um......GEEKS UNITE!

What was the original context of the question? Is this for your GRE studying? In which case it would be a multiple choice question, right?

I think if you're expected to do logs without a calculator then the expectation is probably more about estimating or guess and check.

10^1.447 is somewhere between 10 and 100

10^.5 is a little more than 3 (since 3^2 = 9)

so (10^1)*(10^.5)=(10^1.5) which would be a little over 30
and 10^1.447 would be a little less than that...

but yeah, without the original question in front of me, I'm just guessing on what they're trying to assess from you.
• _gingin

Posts: 116
• FlashDrive

Posts: 53
_gingin said

iguanaSF said

Rawr! That's one of the sexiest things posted on here in years.

People who know me know I'm not kidding.

With gingin's graphical interpretation, we can understand why expansion in math is sexy...
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