Graph natural log of sin(x)

Problem

ln(sin(x))

Solution

1. Identify the domain of the function ƒ(x)=ln(sin(x)) Since the natural logarithm is only defined for positive values, we require sin(x)>0 This occurs on the intervals (2*k*π,(2*k+1)*π) for any integer k

2. Determine the range of the function. The sine function oscillates between 0 and 1 on its valid domain. As sin(x)→0 ln(sin(x))→−∞ When sin(x)=1 ln(1)=0 Thus, the range is (−∞,0]

3. Find the periodicity of the function. Since sin(x) has a period of 2*π the function ƒ(x)=ln(sin(x)) also has a period of 2*π

4. Identify vertical asymptotes where the argument of the log is zero. These occur at x=k*π for any integer k

5. Locate maximum points where the sine function is at its peak. Since the maximum value of sin(x) is 1 at x=π/2+2*k*π the maximum value of the function is ƒ*(π/2+2*k*π)=ln(1)=0

6. Analyze the shape of the graph. The graph consists of a series of downward-opening concave curves separated by gaps where sin(x)≤0 Each curve peaks at y=0 and approaches −∞ at the boundaries of the domain intervals.

Final Answer

ƒ(x)=ln(sin(x))

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