Question

the range of $f(x) = \log_b x$ is the set of all positive real numbers. \
\
a. true \
\
b. false

Explanation:

To determine the range of the logarithmic function \( F(x)=\log_b x \) (where \( b>0, b
eq1 \)):

• Recall the definition and properties of logarithmic functions. The logarithmic function \( y = \log_b x \) is the inverse of the exponential function \( y = b^x \).
• The range of an exponential function \( y = b^x \) (for \( b>0, b

eq1 \)) is the set of all positive real numbers (\( (0,+\infty) \)), and the domain of an exponential function is all real numbers (\( (-\infty,+\infty) \)).

• By the property of inverse functions, the domain of the inverse function (the logarithmic function \( \log_b x \)) is the range of the original function (exponential function), and the range of the inverse function is the domain of the original function.
• So, the domain of \( \log_b x \) is \( (0,+\infty) \) (since it's the range of \( b^x \)), and the range of \( \log_b x \) is \( (-\infty,+\infty) \) (all real numbers), not just positive real numbers.

Answer:

B. False