(reminder: all quotes here are fiddled, probably.)

### 道家, etc. 1. 道家

2. Misconception: The path of a charge and the field lines.

Consider a positive charge moving in an electric field. Does the field line indicate the path it takes? No. The field line only shows the direction of the force acting on the charge, not the direction of the charge's motion. Think of a test charge rotating about a negative one.

How about a charge initially at rest? Will it move along the field line? The answer is no again, except when the charge happens to rest on a straight field line.

3. Log

Which of the following is the smallest?
a. 3^4000
b. 4^3000
c. 6^2000
d. 12^1000

Consider their logarithm.
a => 4 ln 3
b => 6 ln 2
c => 2 ln 2 + 2 ln 3
d => 2 ln 2 + ln 3
Obvious, (d) is the smallest.

Consider f(x) = x^{12/x} , where x is positive.
Noting x ln f(x) = 12 ln x , we get
f'(x) = (12/x - ln f(x)) f(x) / x ,
which equals 0 only when x ln f(x) = 12 ;
i.e. f(x) attains its maximum when x = e .
So, (d) is the smallest (x = 12 , most distant from e).

Further:
* 黎文傑老師: 卷二常考問題
* Solve a=5b and a^{log b}=2