Question

question 1 of 10
which answer represents the range of the logarithmic function given below?
$f(x) = \log_{0.5}x$

a. $y \geq 0$
b. $y < 0$
c. $y > 0$
d. all real numbers

Explanation:

Step1: Recall Logarithmic Function Range

The general form of a logarithmic function is \( y = \log_b x \), where \( b>0, b
eq1 \). The range of any logarithmic function (regardless of the base \( b \), as long as it's a valid logarithmic base) is all real numbers. This is because for any real number \( y \), we can find an \( x \) (specifically \( x = b^y \)) such that \( \log_b x = y \).

Step2: Apply to Given Function

For the function \( F(x)=\log_{0.5}x \), the base \( b = 0.5 \) (which is valid since \( 0.5>0 \) and \( 0.5
eq1 \)). Using the property of logarithmic functions, the range of \( \log_{0.5}x \) should be all real numbers, just like any other logarithmic function.

Answer:

D. All real numbers