I just have a quick question regarding this question:
Simplify. State any restrictions on the variables.
log(x^2+7x+12)-log(x^2-9)
my answer is log[(x+4)/(x-3)] which is correct. However, for the restrictions, I believe that they are: x>3 and x<-4.. But the back of the book says only x>3.
When I plug in x<-4 any number such as x=-5 the expression is defined. So.. my question, is the back of the book wrong? Are there 2 restrictions and not one?
Thanks, please explain your answer fully.
remember you can only take logs of positive numbers.
so in log [(x+4)/(x-3)]
x > -4 AND x > 3,
so clearly x > 3
It seems to me that the restriction x>3 is based on
log[(x+4)/(x-3)] = log(x+4) - log(x-3)
which will not permit x≤3, which also covers x≤-4.
The restriction for the simplification is simply x ≠-3, otherwise we cannot cancel the (x+3) term.
Finally, the function
log[(x+4)/(x-3)]
is defined for
{x ∈ � : x < -4} ∪ {x ∈ � : x > 3}
as you have correctly determined.
To determine the restrictions on the variables, we need to consider the logarithmic expression.
In this case, we have log(x^2+7x+12) - log(x^2-9).
For the expression to be defined, the argument of the logarithmic function (inside the parentheses) must be positive. Additionally, since we are subtracting logarithms, the argument of the second logarithm (x^2-9) cannot be zero.
Let's break this down further:
1. Starting with the first logarithm, x^2+7x+12, we need this expression to be positive:
We can factorize x^2+7x+12 as (x+4)(x+3).
Setting this expression greater than zero: (x+4)(x+3) > 0
To determine the sign of the expression, we can use the sign-test method.
We write all the critical points on the number line: -4 and -3.
We test a value less than -4, e.g., -5: (-5+4)(-5+3) = (-1)(-2) = 2 > 0
We test a value between -4 and -3, e.g., -3.5: (-3.5+4)(-3.5+3) = (0.5)(-0.5) = -0.25 < 0
We test a value greater than -3, e.g., 0: (0+4)(0+3) = (4)(3) = 12 > 0
Putting all the information together, the expression (x+4)(x+3) > 0 when x < -4 or x > -3.
2. Moving on to the second logarithm, x^2-9, we need this expression to be non-zero:
This expression can be factored as (x-3)(x+3).
Setting this expression not equal to zero: (x-3)(x+3) ≠ 0
For this expression to be different from zero, neither (x-3) nor (x+3) can be zero.
Therefore, x cannot be equal to -3 or 3.
So, based on our analysis:
- The first restriction is x < -4 or x > -3, as obtained from the first logarithm.
- The second restriction is x ≠ -3 and x ≠ 3, as obtained from the second logarithm.
Now, to address your concern, let's test x = -5 in the original expression:
log[(-5)^2+7(-5)+12] - log[(-5)^2-9]
= log[25 - 35 + 12] - log[25 - 9]
= log[2] - log[16]
Since each argument in both logarithms is positive, the expression is indeed defined.
Therefore, according to the expression and the restrictions we derived, both x>3 and x<-4 are valid restrictions on the variables. It seems the back of the book may contain an error.