As I have stated before, I'm in pre-calc class, learning easy stuff.
Today, finally something cool happened in the class and arouse my interest.
There is a problem on my homework from orange pre-calc textbook, Precalculus With Limits: A Graphing Approach. Page 214, Problem 85. Solve for x
ln(x+1)²=2
The usual approach
2ln(x+1) = 2
ln(x+1) = 1
x+1 = e
If you did exactly like that then.. you are wrong!
Because there is another approach that find 2 answers
e^ln(x+1)² = e²
e^2ln(x+1) = e²
(e^ln(x+1))² = e²
e^ln(x+1)=±e
x+1 = ±e
How did that happen? how come the rule ln(x^y) = yln(x) doesn't work anymore?
If you read the textbook(I believe, most textbook that teaches logs will say the same thing), it should say that
ln(x^y) = yln(x)
and most importantly, a very NOT noticeable scribble that only 0.00001% percent of the population spots:
x is a positive real number
go back to the original problem, but a bit simplified by taking away the +1
ln(x²)=2
According to the rule, ln(x²) = 2ln(x) only if x is positive.
Think about it, if it works if x<0, you will see this rule also apply.
ln(x²) = ln(x*x) = ln(x) + ln(x)
which would not work because ln(x) is not defined for real numbers when x<0
That's why the other half of the solution is not there.
so, from now on, don't assume that ln(x²) = 2ln(x) unless it states x is not positive.
the more precise one is ln(x²) = 2ln|x|. know how to solve absolute values right? 2ln|x| = 2ln(x) or 2ln(-x), solve each one separately.
If you are going to ask, is ln(x) defined in the complex number system when x<0? yes.
ln(x) = ln(-x) + iπ, x<0.
ln(x^y) = yln(x) still doesn't work for all x even if it's expanded to complex plane.
Proof ln(x) = ln(-x) + iπ
e^ln(x) = e^(ln(-x)(iπ))
x = e^ln(-x)*e^(iπ)
x = -x*-1
x =x
yeahhhhhh...
Actually ln(x) can also be ln(-x) + (2n+1)iπ, where n is a integer, but mathematicians declare it as ln(-x)+iπ for convenient.
I wish there are better textbooks that state this problem. I have read 3 textbooks explaining log, and nothing said something about that. Even Wikipedia's List of logarithmic identities don't have it until I added it up there.
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