Question

question 4 of 10
which facts are true for the graph of the function below? check all that apply.
f(x) = log_{0.521} x

a. it is increasing.

b. the range is all real numbers.

c. the domain is x > 0.

d. the y-intercept is (0, 4).

e. it is decreasing.

f. the x-intercept is (1, 0).

Explanation:

Step1: Recall properties of logarithmic functions

For a logarithmic function \( F(x)=\log_b x \), where \( b>0, b
eq1 \):

• Domain: \( x > 0 \) (since we can only take the logarithm of positive numbers).
• Range: All real numbers (because as \( x \) approaches 0 from the right, \( \log_b x \) approaches \( -\infty \) (if \( b > 1 \)) or \( +\infty \) (if \( 0 < b < 1 \)), and as \( x \) approaches \( +\infty \), \( \log_b x \) approaches \( +\infty \) (if \( b > 1 \)) or \( -\infty \) (if \( 0 < b < 1 \)), covering all real numbers).
• If \( 0 < b < 1 \), the function is decreasing (since the base is between 0 and 1, as \( x \) increases, \( \log_b x \) decreases).
• The \( x \)-intercept is found by setting \( F(x) = 0 \), so \( \log_b x=0\implies x = 1 \) (since \( \log_b 1 = 0 \) for any valid \( b \)). There is no \( y \)-intercept because the domain is \( x>0 \), so \( x = 0 \) is not in the domain.

Step2: Analyze each option

• Option A: The base \( b = 0.521 \), which is between 0 and 1. So the function is decreasing, not increasing. So A is false.
• Option B: For any logarithmic function \( \log_b x \), the range is all real numbers. So B is true.
• Option C: The domain of \( \log_b x \) is \( x>0 \), so C is true.
• Option D: The \( y \)-intercept would be at \( x = 0 \), but \( x = 0 \) is not in the domain of the logarithmic function, so there is no \( y \)-intercept. So D is false.
• Option E: Since the base \( 0 < 0.521 < 1 \), the logarithmic function is decreasing. So E is true.
• Option F: To find the \( x \)-intercept, set \( F(x)=0 \). So \( \log_{0.521}x = 0 \). By the definition of logarithms, \( \log_b x = y\implies b^y=x \). So \( 0.521^0=x\implies x = 1 \). Thus, the \( x \)-intercept is \( (1,0) \), so F is true.

Answer:

B. The range is all real numbers, C. The domain is \( x>0 \), E. It is decreasing, F. The \( x \)-intercept is \( (1, 0) \)