Which of the following logarithms results in a negative number?

To understand why the logarithm in this case results in a negative number, let's examine the expression log base one half of 4.

When calculating logarithms, we essentially ask the question: "To what exponent must the base be raised to obtain a given number?" For log base one half of 4, we are asking, "What power do we raise one half to in order to get 4?"

Since one half is a fraction less than 1, raising it to a positive exponent yields a smaller number, while raising it to a negative exponent yields a larger number. Specifically, when we think about powers of one half:

• When we raise one half to the -n power (for some positive n), we are taking the reciprocal, which results in 2^n.

For this particular logarithm, we find that:

1/2 raised to the power of -2 gives us (1/2)^-2 = 2^2 = 4.

This means that log base one half of 4 equals -2, which is indeed a negative number.

Thus, the logarithm yields a negative result due to the properties of logarithmic functions with bases that are fractions between 0 and 1.