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Old 2007-12-06, 08:16   #1
nibble4bits
 
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Default 2^x using powers of e?

I have an equation like: 2^(xa)=e^x, where a=1/ln (2)
I wonder how I'd go about finding an equation that gives me a power of 2 without using 2^x. In other words, f(g(h(...(x)...))) gives 2^x where it is prefered for f,g,h... to all be in terms of logarithms and powers of e.

I could swear this is a simple matter but I just can't seem to find the correct function. I've been trying to apply the rules I learned in college algerbra a few years ago and can't get it to work. Has anyone got an answer or method to arrive at one?

Thanks, Joe
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Old 2007-12-06, 10:17   #2
retina
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What is wrong with: e^(x*ln(2))=2^x
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Old 2007-12-06, 12:48   #3
xilman
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Quote:
Originally Posted by retina View Post
What is wrong with: e^(x*ln(2))=2^x
"2" appears on the LHS, which may not be acceptable. Try converting your LHS into e^(xln((ln(e)+ln(e)))).

Or, even cleaner, IMO, (ln(e)+ln(e))^x = 2^x


Paul
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Old 2007-12-06, 13:36   #4
R.D. Silverman
 
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Quote:
Originally Posted by nibble4bits View Post
I have an equation like: 2^(xa)=e^x, where a=1/ln (2)
I wonder how I'd go about finding an equation that gives me a power of 2 without using 2^x. In other words, f(g(h(...(x)...))) gives 2^x where it is prefered for f,g,h... to all be in terms of logarithms and powers of e.

I could swear this is a simple matter but I just can't seem to find the correct function. I've been trying to apply the rules I learned in college algerbra a few years ago and can't get it to work. Has anyone got an answer or method to arrive at one?

Thanks, Joe
This isn't college algebra. The fact that you think that it is
illustrates some of the serious problems we have today in mathematics
education. Normally, logarithms are a second year algebra topic for
junior high school (if one isn't clueless), or high school (if one is clueless).
I say this because people who are not clueless generally start taking
algebra BEFORE high school.
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Old 2007-12-06, 15:17   #5
davieddy
 
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Quote:
Originally Posted by R.D. Silverman View Post
This isn't college algebra. The fact that you think that it is
illustrates some of the serious problems we have today in mathematics
education. Normally, logarithms are a second year algebra topic for
junior high school (if one isn't clueless), or high school (if one is clueless).
I say this because people who are not clueless generally start taking
algebra BEFORE high school.
I'm 57 and in danger of losing my sense of humour.
How about you?

David
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Old 2007-12-06, 15:28   #6
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Bob,

In defense of Joe, in my math education, I covered many topics at least three or four times. What we covered at the graduate level encompassed, and extended what we had covered at the undergraduate level. Similarly, the college level included that which was previously covered in high school.

That is not to imply that in the later courses we spent much, if any, time on "how", but probably derived the "why" of a procedure that we learned earlier.

But, if you can really pinpoint just where in your education you first learned each topic, then I fear that you are placing more emphasis on the "educational curriculum" rather than on the "math" itself.

For example, when are the concepts of "set theory" taught?

So, specifically to Joe's remark, I interpret it to be inclusive of "College Algebra" rather than limited to just the additional concepts introduced at that level.
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Old 2007-12-06, 15:42   #7
R.D. Silverman
 
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Quote:
Originally Posted by davieddy View Post
I'm 57 and in danger of losing my sense of humour.
How about you?

David
With respect to education, I lost mine quite some time ago.

And it isn't just in mathematics. My kids attend the SAME high school
that I did. When I compare what I was taught with what they are being
taught, I get disgusted.

Ask the average U.S. high school student today:

Define "dangling particple".
What is "faulty parallel sentence construction"
Where is Ulan Bator located? What about Madagascar? (etc.)
What conflict did Andrew Jackson have with the Supreme Court? How
did he violate the Constitution?
What is electronegativity? Why is it higher for Cesium than Potassium?
What is the electronic configuration for Carbon?
What are Kepler's Three laws? If and object with mass M1 and momentum
P1 collides inelastically with an object of mass M2 and momenturn P2 at
angle theta, describe the vectors of the two objects after collision.
Who said "that which we call a rose, by any other name would smell as
sweet"?

etc. etc. etc.

Kids today are lazy, ignorant, and don't care about educating themselves.
Even my old teachers say the same thing. My generation was much more
educationally motivated than today's generation.

And what I see in this forum is people who just want to run other
people's code without bothering to learn about how/why it works.
They want to pretend they are accomplishing something by factoring
numbers that noone else cares about using code written by others and
about which they are totally ignorant. They want rewards and recognition
without being willing to put in the work. I recall Debbie Allen's (from Fame)
comments: "You want fame? Well fame COSTS."

The wannabees have the right to do what they want. It is their machines
and their time.

And I have the right to be contemptuous.
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Old 2007-12-06, 15:50   #8
davieddy
 
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Quote:
Originally Posted by R.D. Silverman View Post
With respect to education, I lost mine quite some time ago.
Jesus that was quick!
Pink Floyd (We don't need no educashion)

Or was I thinking of "Schools Out":)

Last fiddled with by davieddy on 2007-12-06 at 15:53
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Old 2007-12-06, 16:44   #9
ewmayer
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Quote:
Originally Posted by R.D. Silverman View Post
Ask the average U.S. high school student today:

Define "dangling partic[i]ple".
Back in my wild-n-crazy college days, I almost got arrested for waving my dangling participle at someone. Bob is 100% correct - If I'd been properly taught about that concept in HS, my subsequent legal near-discomfiture could have been avoided.
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Old 2007-12-06, 18:18   #10
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back to the topic...
Quote:
Originally Posted by xilman View Post
"2" appears on the LHS, which may not be acceptable. Try converting your LHS into e^(xln((ln(e)+ln(e)))).

Or, even cleaner, IMO, (ln(e)+ln(e))^x = 2^x
Well, that's basically (1+1)^x, and 2 is by definition 1+1
(independently of how you define +1...)
So another simple solution might be (6/3)^x. Or, exp(ln(6)/ln(3)).
Or exp(ln(6^(1/ln(3))))

well I just notice: "... a power of 2" - does this mean "any" ? Then 1 is the most simple answer. Or 4, if "positive" is tacitely understood.

did I miss something or was there some comment on the first (on-topic) proposals ?


PS: I agree on deficiencies / degradation of the teaching system, but IMHO
this has nothing to do with the average IQ of the kids.
http://en.wikipedia.org/wiki/Cui_bono ?
This is a purely political (I'd even be enclined to say economical) choice.
For politics and economy, it is much better to have stupid consumers.

PPS: and it is quite hard for parents to struggle against the overwhelming influence of that all-encompassing "institution". Probably the absence of a TV set at home would make things easier. Well, starting to struggle already means that parents themselves are not yet brainwashed. Gets rare, that species...

Last fiddled with by m_f_h on 2007-12-06 at 18:26
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Old 2007-12-06, 18:36   #11
R.D. Silverman
 
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Quote:
Originally Posted by m_f_h View Post
back to the topic...



PS: I agree on deficiencies / degradation of the teaching system, but IMHO
this has nothing to do with the average IQ of the kids.
http://en.wikipedia.org/wiki/Cui_bono ?
This is a purely political (I'd even be enclined to say economical) choice.
For politics and economy, it is much better to have stupid consumers.
I have a piece of anecdotal evidence that supports my views
about the dumbing down of America.

When I was in high school, being selected into the National Honor
Society was (somewhat of) an honor. From my class of 430, we had
about 35 students selected. It was considered the equivalent of
Phi Beta Kappa for high school.

My daughter was just inducted. Now, it is no longer considered the
equivalent to Phi Beta Kappa. To join, one is also selected for
"leadership, and "performing community service". Indeed, one needs to
get documents signed showing that one has performed some minimal
number of hours of community service. Not only that, but the
academic requirements have been watered down. To qualify, one needs
an overall overage of 98/100 where the average is computed by giving
a 20% extra weight to any honors course and a 15% weight to the level
of courses just below honors level. YECH!!!!

Some 122 students in my daughter's class were elected. This is about
28% of her class. WHAT A JOKE!!!! This "honor society" has become
about as exclusive as a rainstorm. The "honor" has been so totally diluted
as to be almost meaningless.

If one also looks at the "academic honor roll" for the high school that is
printed in the local paper, and compares the number of students getting
"high honors" vs. my high school class, it is 3 to 4 times as many students.
So grade inflation has struck as well.

Now, if students were actually getting smarter, or learning more, this
could all be justified. I note, however, that the number of National
Merit scholars has actually DECLINED since I went to high school.
My class had 8 finalists and some 19 honorable mentions. (Last year's
class only had 3 finalist and half the number of honorable mentions)
(For non-U.S. people, the National Merit Scholarships are awarded
as the result of a competitive test in Math & English. They are the
preliminary SAT tests)

It is all part of the dumbing down of America.
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