To find the exact value of the logarithmic expression [tex]\(\log_7(-7)\)[/tex], we need to consider the fundamental properties of logarithms and the number system in which we are working.
1. Understanding Logarithms: The logarithm [tex]\(\log_b(a)\)[/tex] answers the question: "To what power must the base [tex]\(b\)[/tex] be raised, to produce the number [tex]\(a\)[/tex]?"
2. Base and Argument Constraints: In the real number system, logarithms are only defined for positive bases (other than 1) and positive arguments. That is, both [tex]\(b\)[/tex] and [tex]\(a\)[/tex] must be positive. A negative number cannot be obtained by raising a positive number to any real power.
3. Applying the Constraints:
- Here, our base [tex]\(b\)[/tex] is 7, which is positive and valid.
- However, our argument [tex]\(a\)[/tex] is -7, which is negative.
4. Undefined Result: Since you are trying to take the logarithm of a negative number, [tex]\(\log_7(-7)\)[/tex] is undefined in the real number system.
Therefore, the logarithmic expression [tex]\(\log_7(-7)\)[/tex] does not have a value in the real numbers and is considered undefined. Hence, there is no real number solution for this expression.
