Graph natural log of cos(x)

Problem

ln(cos(x))

Solution

1. Identify the domain of the function. The natural logarithm ln(u) is only defined for u>0 Therefore, we must have cos(x)>0 This occurs in the intervals (−π/2+2*k*π,π/2+2*k*π) for any integer k

2. Determine the range of the function. Since the maximum value of cos(x) is 1 the maximum value of the function is ln(1)=0 As cos(x) approaches 0 from the right, ln(cos(x)) approaches −∞ Thus, the range is (−∞,0]

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

4. Identify vertical asymptotes. The function approaches −∞ as cos(x) approaches 0 This happens at x=π/2+k*π for any integer k

5. Find the derivatives to determine the shape. The first derivative is:

d(ln(cos(x)))/d(x)=(−sin(x))/cos(x)=−tan(x)

The second derivative is:

(d(−)*tan(x))/d(x)=−sec2(x)

Since the second derivative is always negative where the function is defined, the graph is always concave down.

6. Sketch the graph. The graph consists of a series of concave-down curves. Each curve has a local maximum of 0 at x=2*k*π and drops toward −∞ at the vertical asymptotes x=±π/2,±(3*π)/2,…

Final Answer

y=ln(cos(x))

Want more problems? Check here!