Explain why the logarithm of a negative number is undefined in the real numbers.
The correct answer and explanation is:
The logarithm of a negative number is undefined in the real number system because logarithms are based on exponential functions, which only produce positive results.
To understand this, recall that the logarithmic function is the inverse of the exponential function. In simpler terms, for any positive base b>0b > 0, the logarithm logb(x)\log_b(x) answers the question: “To what power must the base bb be raised to produce xx?” Mathematically, if y=logb(x)y = \log_b(x), then by=xb^y = x.
Now, when the base b>1b > 1, the exponential function byb^y always gives a positive value for any real value of yy. For instance, 2y2^y will only produce positive results, regardless of whether yy is positive, negative, or zero. Therefore, there is no real number yy that can satisfy the equation by=−xb^y = -x, where xx is a positive number, because the exponential function cannot yield negative numbers. Thus, it is impossible to take the logarithm of a negative number in the real number system.
In contrast, in the complex number system, logarithms of negative numbers can be defined using Euler’s formula, which connects exponentials with complex numbers. The complex logarithm can be written as: logb(x)=logb(∣x∣)+i(arg(x)+2kπ)\log_b(x) = \log_b(|x|) + i(\arg(x) + 2k\pi)
where ∣x∣|x| is the absolute value of xx, and arg(x)\arg(x) is the argument of xx. This allows logarithms of negative numbers to be defined in the complex plane, but in the real numbers, such logarithms are undefined.