To solve for [tex]\(g(-2)\)[/tex] given the equation [tex]\(3 \log(x) = x^2 + 7\)[/tex] and the function [tex]\(g(x) = 3 \log(x) - (x^2 + 7)\)[/tex], follow these steps:
1. Understand the given function [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = 3 \log(x) - (x^2 + 7) \][/tex]
2. Substitute [tex]\(x = -2\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[ g(-2) = 3 \log(-2) - ((-2)^2 + 7) \][/tex]
3. Evaluate inside the logarithm function [tex]\(\log(-2)\)[/tex]:
The logarithm function [tex]\(\log(x)\)[/tex] (usually the common log base 10) is defined only for positive values of [tex]\(x\)[/tex]. Since [tex]\(\log(-2)\)[/tex] is not defined in the real numbers, it indicates that [tex]\(g(-2)\)[/tex] is undefined in the real number system.
Therefore, the value of [tex]\(g(-2)\)[/tex] is undefined because the logarithm of a negative number does not exist within the real numbers.
