confused when to ignore negative solutions

[Nomvalt]

Solve for x in the following.
log₃(x) + log₃(x+2) = log₃(
log₃(x²+2x) = log₃(
x² + 2x - 8 = 0
(x+4)(x-2) = 0
x= -4 or x = 2

Though why is the answer only supposed to be x = 2? Why not x = -4 also? Does it being a negative value have something to do with it? I thought you only ignore the value of x if it's a negative number and you are trying to find the unknown value of the base of a logarithmic equation. Here we are trying to determine the number component in the logarithmic equation.

Solve for x in the following.
(log₁₀(x))² + log₁₀(x) - 2 = 0

Letting a = log₁₀(x)
a² + a - 2 = 0
(a+2)(a-1)=0
a=1 or a=-2

log₁₀(x) = 1             log₁₀(x) = -2
x=10                                      x=1/100 

And here why is the answer x=10 and x=1/100? Why isn't the answer just x=10 as only a=1 is positive.
In a nutshell I'm confused about when we are supposed to ignore a negative solution. Would anyone be able to provide explanations for the above?

[stonecold]

sub your answers back into the question.  the ones that give negatives are invalid, as you can't have a negative or zero inside a log...

[Yitzi_K]

The reason it can't be a negative is because you can't have a negative inside a log function.

Think about it: y=log_3_(x) means 3^y=x. There is no value of y which could give a negative result. Any positive number to the power of anything will always give a positive result. (Disregarding complex numbers).

[m@tty]

For a generic logarithm, its domain is , so a negative number will render it undefined.

Similarly a second logarithm, has a domain of . So any x value less than negative c is a 'false' solution, because it is undefined at that point.

For an equation involving a sum of logarithms, like your example above, every one of the logarithms must be define at a certain value of x for that value to be included in the domain.

In your example the domain is so the domain is simply

So any solution you determine must be checked against this domain, if it is not within it then it is not a valid solution.

Basically, as stonecold said, sub your values back in to the original expression and see if it is defined.

[qshyrn]

a log of something positive CAN give you a negative, but u CANT log something negative. keep that in mind and ull be fine

so thats why logx could be -2. and in the first one u cant have log(-4) as 10 to the power of something will never be negative

[happyhappyland]

Just think of a log graph, if you cant graph it it cant be done.