True or False? In Exercises 97-100 , determine whether the...

True or False? In Exercises $97-100$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$$\ln \left|\cos \theta^{2}\right|=\ln \left(\cos \theta^{2}\right)$$

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Key Concepts

Absolute Value Function

The absolute value function converts any real number to its nonnegative magnitude. When applied inside a logarithm, it guarantees that the argument is nonnegative; however, it does not change the value for inputs that are already positive. Understanding how absolute values interact with other functions like logarithms helps in determining whether certain algebraic manipulations or equalities are valid.

Equality Conditions Involving Logarithms and Absolute Values

An equality of the form ln|x| = ln(x) holds if and only if x is positive. This condition is essential because the logarithm of a negative number is undefined while the logarithm of the absolute value is defined. Recognizing when one can remove the absolute value sign without changing the domain or altering the expression is a key aspect of analyzing such statements.

Logarithm Function Domain

The logarithm function, particularly ln, is defined only for positive arguments. This means that when taking the logarithm of any expression, one must ensure that the expression inside the logarithm is strictly greater than zero. If an expression can be negative, the logarithm is not defined for those values, which is why domain considerations are a critical part of solving such problems.

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