XPost: sci.lang, alt.usage.english

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3


https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......


i wonder if there's a way to get better (and better) approximations.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english

HenHanna <HenHanna@dev.null> writes:

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3

https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......

i wonder if there's a way to get better (and better) approximations.

Is it possibly you can summarize he gist of it in here so we can
discuss?

Did you take calculus in college and if so, did you ever learn the limit definition of the derivative?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english

On Mon, 18 Nov 2024 14:51:15 +0000, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3


https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......


i wonder if there's a way to get better (and better) approximations.

Is it possibly you can summarize he gist of it in here so we can
discuss?

Did you take calculus in college and if so, did you ever learn the limit definition of the derivative?

i thnk... i knew the Epsilon-Delta def. when i was 13.

Log2(3) = x

so 2^x =3 ---------- Square both sides

2^(2x) = 9 ------ We know that 2^3 = 8

2^(2x) > 2^3 ---- (2^power is monotonic) (monotonically increasing)

2x > 3

x > 1.5


We get x < 1.6666...... by Cubing both sides


i wonder if there's a way to get better (and better) approximations.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english, sci.math

HenHanna <HenHanna@dev.null> writes:

On Mon, 18 Nov 2024 14:51:15 +0000, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3


https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......


i wonder if there's a way to get better (and better) approximations.

Is it possibly you can summarize he gist of it in here so we can
discuss?

Did you take calculus in college and if so, did you ever learn the limit
definition of the derivative?

i thnk... i knew the Epsilon-Delta def. when i was 13.

Log2(3) = x

so 2^x =3 ---------- Square both sides

2^(2x) = 9 ------ We know that 2^3 = 8

2^(2x) > 2^3 ---- (2^power is monotonic) (monotonically increasing)

I've been out of college for twenty years and, even though I studied
math, there's more rust than anything. I see you chose the closest cube
from 9 to achieve a clean cube root. Which operation did you
do to get 2x > 3? Did you log both sides? Don't hit me if that's a
stupid question.

2x > 3

x > 1.5

We get x < 1.6666...... by Cubing both sides

How do achieve a result of 1.666666 by cubing 1.5? I get 1.5^3 = 3.375.

i wonder if there's a way to get better (and better) approximations.

Ever visit sci.math? I'm in there, perhaps we could crosspost this into
that NG and include them in the conversation. Oh, I will.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english

In article <97e7e6fb078310c8d4d600c247847957@www.novabbs.com>,
HenHanna <HenHanna@dev.null> wrote:

i wonder if there's a way to get better (and better) approximations.

Look for more powers of 2 near to powers of 3.

For example,

3^7 (= 2187) > 2^11 (= 2048), so 3 > 2^(11/7), so log2(3) > 11/7 = 1.571+ 3^10 (= 59049) < 2^16 (= 65536), so 3 < 2^(16/10), so log2(3) < 10/6 = 1.6

3^12 is very close to 2^19, so log2(3) is very close to 19/12 = 1.583+

-- Richard

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english

On Tue, 19 Nov 2024 19:47:00 +0000, Richard Tobin wrote:

In article <97e7e6fb078310c8d4d600c247847957@www.novabbs.com>,
HenHanna <HenHanna@dev.null> wrote:

i wonder if there's a way to get better (and better) approximations.

Look for more powers of 2 near to powers of 3.

For example,

3^7 (= 2187) > 2^11 (= 2048), so 3 > 2^(11/7), so log2(3) > 11/7 =
1.571+
3^10 (= 59049) < 2^16 (= 65536), so 3 < 2^(16/10), so log2(3) < 10/6 =
1.6

3^12 is very close to 2^19, so log2(3) is very close to 19/12 = 1.583+

-- Richard

Look for more powers of 2 near to powers of 3.

Thank you.... is there a good way to look for them?


___________

Why didn't i tihnk of that???

i saw a really nice Clip on Youtube.
(i think from the 3-Brown 1-Blue guy)

Plot all points (P cis P) for all prime P

and the result was Spirals. (amazing graphics effect Zooming out and
in)


The spirals resulted from a similar idea (or phenomenon) of
when multiples of P is close to multiples of 2 Pi.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english

On Mon, 18 Nov 2024 22:03:24 +0000, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

On Mon, 18 Nov 2024 14:51:15 +0000, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3


https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......


i wonder if there's a way to get better (and better) approximations. >>>

Is it possibly you can summarize he gist of it in here so we can
discuss?

Did you take calculus in college and if so, did you ever learn the limit >>> definition of the derivative?

i thnk... i knew the Epsilon-Delta def. when i was 13.

Log2(3) = x

so 2^x =3 ---------- Square both sides

2^(2x) = 9 ------ We know that 2^3 = 8

2^(2x) > 2^3 ---- (2^power is monotonic) (monotonically increasing)

I've been out of college for twenty years and, even though I studied
math, there's more rust than anything. I see you chose the closest cube
from 9 to achieve a clean cube root. Which operation did you
do to get 2x > 3? Did you log both sides? Don't hit me if that's a
stupid question.

2x > 3

x > 1.5


We get x < 1.6666...... by Cubing both sides

How do achieve a result of 1.666666 by cubing 1.5? I get 1.5^3 = 3.375.

i wonder if there's a way to get better (and better) approximations.

Ever visit sci.math? I'm in there, perhaps we could crosspost this into
that NG and include them in the conversation. Oh, I will.

Thanks.

so 2^x =3 ---------- Square both sides

so 2^x =3 ---------- Cubing both sides

2^(3x) = 27 ------ We know that 2^5 = 32

............. we get x < 1.6666......


________________ from this NewsReader i can post (max 3 groups)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: alt.usage.english

In article <1c945d5b40bb60fb5a27dca42ea9629f@www.novabbs.com>,
jerryfriedman <jerry.friedman99@gmail.com> wrote:

Good to see you in a.u.e., Richard!

Oops! I didn't notice the cross-posting from rec.puzzles.

And thanks. Perhaps I should look in again.

-- Richard

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.lang, alt.usage.english, sci.math

On 11/18/24 5:03 PM, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

On Mon, 18 Nov 2024 14:51:15 +0000, Daniel wrote:

HenHanna <HenHanna@dev.null> writes:

is there a good way to get this value by hand?

(without a Calculator) Log (base 2) of 3


https://www.youtube.com/watch?v=X6C5hGpWW5A

This clip shows how to derive

1.5 < Log2(3) < 1.6666666......


i wonder if there's a way to get better (and better) approximations. >>>

Is it possibly you can summarize he gist of it in here so we can
discuss?

Did you take calculus in college and if so, did you ever learn the limit >>> definition of the derivative?

i thnk... i knew the Epsilon-Delta def. when i was 13.

Log2(3) = x

so 2^x =3 ---------- Square both sides

2^(2x) = 9 ------ We know that 2^3 = 8

2^(2x) > 2^3 ---- (2^power is monotonic) (monotonically increasing)

I've been out of college for twenty years and, even though I studied
math, there's more rust than anything. I see you chose the closest cube
from 9 to achieve a clean cube root. Which operation did you
do to get 2x > 3? Did you log both sides? Don't hit me if that's a
stupid question.

2x > 3

x > 1.5


We get x < 1.6666...... by Cubing both sides

How do achieve a result of 1.666666 by cubing 1.5? I get 1.5^3 = 3.375.

i wonder if there's a way to get better (and better) approximations.

Ever visit sci.math? I'm in there, perhaps we could crosspost this into
that NG and include them in the conversation. Oh, I will.

So, you have 1.5 < log2(3) < 1.666

Take 2 to the power of each side since that is monotonic

2^1.5 < 3 < 2^1.666


For 2^1.5 < 3, square both sides and get 2^3 < 3^2 8 < 9

for 3 < 2^1.6666 cube both sides, 3^3 < 2^5 27 < 32

So, we can find the approximations by finding relationships between
powers of 3 and powers of 2

If 3^n < 2^m then log2(3) < m/n
and if 2^m < 3^n then m/n < log2(3)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Tue, 19 Nov 2024 19:47:00 +0000 (UTC), Richard Tobin wrote:

In article <97e7e6fb078310c8d4d600c247847957@www.novabbs.com>,
HenHanna <HenHanna@dev.null> wrote:

i wonder if there's a way to get better (and better) approximations.

Look for more powers of 2 near to powers of 3.

For example,

3^7 (= 2187) > 2^11 (= 2048), so 3 > 2^(11/7), so log2(3) > 11/7 =
1.571+
3^10 (= 59049) < 2^16 (= 65536), so 3 < 2^(16/10), so log2(3) < 10/6 =
1.6

3^12 is very close to 2^19, so log2(3) is very close to 19/12 = 1.583+

-- Richard

That's a very nice explanation.

Not a direct answer to HenHanna's original question, but interesting none
the less. I was trying to work out how Babbage's difference engine, using finite differences, could be used to perform relaterd calculations.I got a
bit distracted, but the following translation of Briggs' ARITHMETICA LOGARITHMICA was very informative.

https://www.17centurymaths.com/contents/albriggs.html



--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)