For what values of x is the expression log(x^2 - 4x- 5 + 2x)/sqrt(x - 1) defined? Express your answer in interval notation.

sandwich Sep 25, 2023

#1**0 **

To determine the values of x for which the given expression is defined, we need to consider two aspects:

The argument of the logarithm (logarithm domain).

The denominator involving the square root.

Let's address each aspect separately:

Logarithm Domain:

The argument of a logarithm must be greater than zero for the logarithm to be defined. In this case, we have the argument as:

x^2 - 4x - 5 + 2x

We need to find when this expression is greater than zero:

x^2 - 4x - 5 + 2x > 0

Simplify the inequality:

x^2 - 2x - 5 > 0

Now, we can find the values of x that satisfy this inequality. You can use methods like factoring or the quadratic formula, but I'll provide the factorized form:

(x - 3)(x + 1) > 0

To determine the sign of the expression on the left-hand side, we can create a sign chart by considering the signs of each factor:

(x - 3) is positive for x > 3 and negative for x < 3.

(x + 1) is positive for x > -1 and negative for x < -1.

Now, we need the product of these factors to be greater than zero, which means both factors must have the same sign. This occurs when:

x > 3 and x > -1 (since both factors are positive).

Therefore, for the logarithm to be defined, x must be greater than 3.

Square Root Denominator:

The denominator of the square root is:

sqrt(x - 1)

For the denominator to be defined, the expression inside the square root (x - 1) must be greater than or equal to zero:

x - 1 >= 0

Solve for x:

x >= 1

Now, we have two conditions:

For the logarithm to be defined: x > 3.

For the denominator (square root) to be defined: x >= 1.

To find the values of x for which the entire expression is defined, we consider the intersection of these two conditions:

x > 3 (logarithm condition) and x >= 1 (denominator condition).

The values of x that satisfy both conditions are x >= 3. Therefore, the expression is defined for all x in the interval [3, ∞).

Saimed Sep 26, 2023