Hi there
I need help with the restrictions on the variables of this question:
Simplify. State any restrictions on the variables.
log(x^2+7x+12)/log(x^2-9)
So my answer is: log(x+4/x-3) which is correct.
Now for the restrictions, I have:
x<-4 and x>3
However the back of the book says the answer is only x>3 and not x<-4 for the restrictions of the variables. But when I sub in any number such as -5 into the ORIGINAL log equation, i get a defined and positive log of x value. So? is mine correct or theirs? Explain fully.
Sorry the question is:
log[(x^2+7x+12)/x^2-9)]
first of all
log(x^2+7x+12)/log(x^2-9) is NOT equal to
log((x+4)/(x-3)) like you say
If the back of your book has that answer, then they are definitely WRONG
log [(x^2+7x+12)/(x^2-9)]
would be equal to log((x+4)/(x-3))
to find the restrictions you have to consider the numerator and denominator
remember we can only take logs of positive numbers
at the top:
log(x^2+7x+12) = log[(x+3)(x+4)]
x < -4 OR x > -3
at the bottom:
log(x^2 - 9) = log[(x-3)(x+3)]
x < -3 or x > 3
but both of these conditions must be met, so
x < -4 OR x > -3 AND x < -3 or x > 3
which leaves you with x < -4 or x > 3
notice that between -4 and +3 we get a negative for either one or the other function
another way to understand the restriction would be:
graph both y = (x+3)(x+4) and
y = (x+3)(x-3)
any x where either or both of those graphs dip below the x-axis would not be allowed.
so yes, they are wrong with their restrictions.
ARGHHHH! saw your correction only after I posted.
So my answer would match your first posting, and it would clearly be a more interesting question.
so now
log [(x+4)(x+3)/((x-3)(x+3))]
= log(x+4) + log(x+3) - log(x+3) - log(x-3)
now each of those terms must be defined.
so x>-4 AND x>-3 AND x>3
which is x > 3
notice that we canceled out log(x+3)
but for all x โค -3 that term would be undefined. So can we really do algebra with undefined numbers??
I know this is a bit tricky, since -5 (or any x < -4) would give a positive result in the original algebraic expression, but it would not work in the individuals terms.
To determine the correct answer for the restrictions on the variables, let's review how to simplify and evaluate logarithms properly.
In the given expression, log(x^2 + 7x + 12)/log(x^2 - 9), the restrictions on the variables depend on the domain and range of the logarithmic functions involved.
First, let's simplify the expression. The numerator, log(x^2 + 7x + 12), can be further simplified as log((x + 3)(x + 4)), while the denominator, log(x^2 - 9), can be simplified as log((x - 3)(x + 3)).
Now, the restrictions on the variables occur when the logarithmic functions are undefined. A logarithmic function is undefined for non-positive arguments. Thus, both the numerator and the denominator must be greater than zero to have a valid solution. Let's analyze each separately:
1. Numerator:
For log(x^2 + 7x + 12) to be defined and positive, the argument (x + 3)(x + 4) must be greater than zero:
(x + 3)(x + 4) > 0
To find the intervals that satisfy this inequality, we use the concept of signs and intervals. We evaluate the sign of the expression (x + 3)(x + 4) for different intervals of x:
Interval Sign
------------------------------------
x < -4 -
-4 < x < -3 +
-3 < x <-4 -
x > -3 +
From the sign analysis, we can determine that (x + 3)(x + 4) > 0 when x < -4 or x > -3.
2. Denominator:
For log(x^2 - 9) to be defined and positive, the argument (x - 3)(x + 3) must be greater than zero:
(x - 3)(x + 3) > 0
Similarly, using sign analysis, we find that (x - 3)(x + 3) > 0 for x < -3 or x > 3.
To obtain the restrictions on x for the original expression, we need the intersection of the restrictions for the numerator and the denominator. Therefore, the correct restrictions on the variables are x > 3.
Regarding the discrepancy between your answer (x < -4 and x > 3) and the answer in the book (x > 3), the reason is that the restrictions are not directly related to the argument of the original logarithm but to the individual logarithmic functions, which can have different domains and ranges.
It's important to note that when you substitute a value like -5 into the original log equation, you are evaluating the expression, but it doesn't guarantee that the logarithm will be positive or defined. To determine the restrictions, we must consider the properties and domains of logarithmic functions.
In conclusion, the correct restrictions on the variables for the given expression are x > 3.