非常有價值的問題（關於對數方程的解）(1)

2008-12-30 1:19 pm

從一條中四至中五程度的logarithm equation開始說起：
log (3x + 4) + log (x + 1) = 1
log [(3x + 4)(x + 1)] = log 10
(3x + 4)(x + 1) = 10
3x2 + 3x + 4x + 4 = 10
3x2 + 7x – 6 = 0
(3x – 2)(x + 3) = 0
x = 2/3 or x = – 3

會考程度的教科書，甚至是數學科老師，一定會教你reject「x = – 3」這個解，其原因是類似「沒有負數的logarithm」之類的解釋。

可是，這個世界上是有complex logarithm的。不信的話請看
http://en.wikipedia.org/wiki/Complex_logarithm、
http://mathworld.wolfram.com/NaturalLogarithm.html和
http://mathworld.wolfram.com/CommonLogarithm.html。

而complex logarithm正好是擴展至所有「complex number的logarithm」（當然亦包括所有「負數的logarithm」）。

因此類似「沒有負數的logarithm」之類的所謂「解釋」就reject某個（些）解其實只是因為「complex logarithm不在中學課程範圍」而出現的「藉口」。

所以現在請你們拋開「會考程度教科書」、甚至是「數學科老師」的成見，面對complex logarithm，判別x = – 3是否符合方程log (3x + 4) + log (x + 1) = 1。

回答 (1)

Jacob

2008-12-30 8:12 pm

✔ 最佳答案

For complex logarithm, the formula is like that:
ln z = ln r + i(θ+2kπ), where k is any integer.
That means complex log is a multi-valued function.

First, ln M+ln N = ln(MN) is NOT valid again for a particular k.
However, it is true for a general sense,
the equality holds in the sense that the whole set is equal.

To see this, take M=N=-1, then for k=0,
ln M+ln N = iπ+iπ = 2πi
ln(MN) = ln 1 = 0

But for the whole set of k, ln M+ln N = ln(MN) is true.

Also, log 10 = 1 is only true for real number,
for complex log, it is not completely true,
in the sense that it is valid for only a particular k.

To see this, for simplicity, consider ln e.
ln e (complex)= ln e (real)+ 2kπi = 1+2kπi.
Thus in complex sense, ln e is NOT same as 1.

Therefore, your first step is NOT valid in complex log.

Lastly, to answer your question, putting x=-3,
LHS = log(-5)+log(-2).

Now compute
log(-5) = log 5(real) + i(2k+1)π/ln10 where k is an integer,
log(-2) = log 2(real) + i(2m+1)π/ln10 where m is an integer.

Thus
LHS = log(-5) + log(-2)
= [log 5 + log 2](real) + 2πi[k+m+1]/ln10
= 1 + 2πi[k+m+1]/ln10

If you want LHS = RHS = 1 is the real sense,
you must setting k+m+1=0, thus you have k+m=-1.

Therefore for complex log, x=-3 is the root
only in the condition that k+m=-1. Otherwise not.

參考: ME

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