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

[Daniel]

is there a good way to get this value by hand?
(without a Calculator) Log (base 2) of 3

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.

[Richard Damon]

Post by Daniel

is there a good way to get this value by hand?
(without a Calculator) Log (base 2) of 3
http://youtu.be/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)