Step 1: Understanding the Concept:
To find the range of a composite function \( y = f(g(x)) \), we first determine the range of the inner function \( g(x) \).
Then, we see what outputs are generated when those values are used as inputs for the outer function \( f(x) \).
Here, \( g(x) = \sin x \) and \( f(x) = \log x \).
We must also satisfy the given domain constraint \( \sin x>0 \).
Step 2: Key Formula or Approach:
1. Range of \( \sin x \): Always between -1 and 1.
2. Logarithmic behavior: \( \log(t) \) increases monotonically. As \( t \to 0^+ \), \( \log(t) \to -\infty \). At \( t=1 \), \( \log(t) = 0 \).
Step 3: Detailed Explanation:
The inner function is \( \sin x \).
By standard definition, \( -1 \leq \sin x \leq 1 \).
The problem specifies the domain condition \( \sin x>0 \).
Therefore, the possible values for the inner argument are \( (0, 1] \).
Now, let's map these values through the natural log function \( y = \log(\sin x) \):
- When \( \sin x = 1 \) (maximum possible value), \( y = \log(1) = 0 \).
- As \( \sin x \) gets smaller and smaller, approaching 0 from the positive side (\( \sin x \to 0^+ \)), the log function goes to negative infinity: \( y = \log(\sin x) \to -\infty \).
- Since the sine function is continuous and hits all values between 0 and 1, the logarithm function hits all corresponding values from \( -\infty \) up to 0.
- Thus, the range consists of all non-positive real numbers.
- In interval notation, this is written as \( (-\infty, 0] \).
Step 4: Final Answer:
The range of the function is \( (-\infty, 0] \).