Help in checking Logs... (1 Viewer)

[noobonastick]

hi I have a rather simple question...

key:

_ = sub
^= to the power of

2 log_4 (x+2) = 0

so i solved for x, which is x= -1 and x=-3 (rather simple) and i was wondering on the method to check the answer

so i sub x = -1 into the equation and it works out.

i sub x=-3 into the equation, and apparently its not an answer.
since its 2.log_4 (-1) (not equal to) 0. Why cant you just move the 2 in front of the equation and square -1, making the equation log_4(1) = 0 and therefore x=-3 is a valid solution.

And lastly i would like to know when do you have to check a logarithm..? when solving for x and theres 2 answers?

thanks =]

 

[Trebla]

You can only take the log of positive numbers. So x + 2 > 0 => x > -2.
As soon as you put the 2 as an index within the logarithm, you have altered the domain to all x except x = -2
log_a(x)ⁿ = nlog_a(x) is only true if x > 0
So,
log₄(x + 2)² = 2log₄(x + 2) is only true for x > -2. It is NOT true otherwise.

So when solving:
2log₄(x + 2) = 0
You need to acknowledge that x > -2, thus x = -1 only, regardless of what method you do.
If you convert it to:
log₄(x + 2)² = 0 ***
This conversion is correct ONLY if x > -2
(x + 2)² = 1
x = -3, - 1
But since the above *** is only true for x > -2, then x = -1 only

Note that solving a question given as:
2log₄(x + 2) = 0
is DIFFERENT to solving another question as:
log₄(x + 2)² = 0

The first one has a domain of x > -2
The seond one has a domain of all real x except x = -2

 

[behemoth100]

Trebla is absolutely correct of course (thats a uni grade answer ) but don't sweat it if you find the proof/rason hard to follow, that is a concept beyond 2 unit and I doubt they would ask you anything along these lines even in 4 unit.

This question is not so much asking you to look at the function itself but, as T said, the domain. f(x)= x^2 for x=>0 is not the same as f(x) = x^2 for all x. Don't believe me ? Sure if you represented your f(x) as a set you would have [0,infinite) but graphically they look every different. Its important to know in Maths that the domain defines the function as much as the function defines the function.
The codomain (range basically, bot not the precise definition) also is important: sin(x) has an inverse if the codomain is [-1,1] but no inverse if the codomain is all numbers (try sticking shift sin 1.5 into your calculator. What do you get).

So with your qs, 2log(x+2) is actually different to log(x+2)^2 graphically, because the domains are different.

But like I said this is a concept that is far beyond year 12 (and one that, if T hadn't pointed out the domain change, I may have gotten wrong in a uni exam).

So its info thats good to have and think about, but that shouldn't freak you out