Graph cos( natural log of x)

Problem

cos(ln(x))

Solution

1. Identify the domain of the function. Since the natural logarithm ln(x) is only defined for positive values, the domain is x>0

2. Analyze the behavior as x→∞ As x increases, ln(x) grows slowly toward infinity. The cosine function will oscillate between −1 and 1 with increasing wavelengths.

3. Analyze the behavior as x→0 As x approaches zero from the right, ln(x) approaches −∞ This causes the function to oscillate infinitely fast between −1 and 1 as it nears the y-axis.

4. Find the x-intercepts. The function equals zero when ln(x)=π/2+n*π for any integer n Solving for x gives x=e(π/2+n*π)

5. Find the local extrema. The function reaches its maximum of 1 when ln(x)=2*n*π or x=e(2*n*π) It reaches its minimum of −1 when ln(x)=(2*n+1)*π or x=e((2*n+1)*π)

6. Sketch the graph. Start from the right where the oscillations are wide and slow, then compress the oscillations infinitely as you move left toward x=0

Final Answer

y=cos(ln(x))

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