1. ## Quadratic involving logarithm problem?

Hi all

I can see here, that

$8n^2 = 64nlog_2n$

can be arranged into $8n^2 - 64nlog_2n = 0$ (if I'm correct on that), but I'm just not sure where to go from here.

If I'm wrong, then I tried diving by 64n on each side to get

$n/8 = log_2n$,

then applying one of the log laws on each side to get

$2^n^/^8 = 2^l^o^g^_^2^n$

Finally: $2^n^/^8 = n$

Then got stuck here. =(

2. Originally Posted by RossR
Hi all

I can see here, that

$8n^2 = 64nlog_2n$

can be arranged into $8n^2 - 64nlog_2n = 0$ (if I'm correct on that), but I'm just not sure where to go from here.

If I'm wrong, then I tried diving by 64n on each side to get

$n/8 = log_2n$,

then applying one of the log laws on each side to get

$2^n^/^8 = 2^l^o^g^_^2^n$

Finally: $2^n^/^8 = n$

Then got stuck here. =(
All your considerations and calculations are OK.

You can't solve the last equation algebraically. So use a numerical method (for instance Newton) to get an approximate result.

I've got x = 1.099997030

3. Thanks! I am interested to know, how you knew this could not be solved algebraically, I guess during your studies numerical methods? Did you study this at university? Is this a topic they went over in your school before uni?

4. Originally Posted by RossR
Thanks! I am interested to know, how you knew this could not be solved algebraically, I guess during your studies numerical methods? Did you study this at university? Is this a topic they went over in your school before uni?
Equations involving functions of mixed type (eg. polynomial and transcendental) can rarely be solved using algebraic methods (unless new functions such as the Lambert W-function are recruited).

5. Well, not all that "new". The W function was defined by Johann Lambert in the 1760s, about 250 years ago!

6. Originally Posted by HallsofIvy
Well, not all that "new". The W function was defined by Johann Lambert in the 1760s, about 250 years ago!
In terms of the age of much of the mathematics studied by a typical maths student, 250 years ago is quite recent! lol!