Graph sin( natural log of x)

Problem

sin(ln(x))

Solution

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

2. Determine the zeros of the function. The sine function is zero when its argument is an integer multiple of π

ln(x)=n*π

x=e(n*π)

where n is any integer.

3. Analyze the behavior as x→0 As x approaches zero from the right, ln(x) approaches −∞ Because the sine function oscillates between −1 and 1 as its argument goes to −∞ the function sin(ln(x)) oscillates infinitely fast as x approaches 0

4. Analyze the behavior as x→∞ As x increases, ln(x) grows very slowly toward ∞ The function will continue to oscillate between −1 and 1 but the distance between the peaks and troughs (the wavelength) increases exponentially.

5. Find the extrema by setting the derivative to zero.

d(sin(ln(x)))/d(x)=cos(ln(x))⋅1/x

The derivative is zero when cos(ln(x))=0 which occurs at ln(x)=π/2+n*π

x=e(π/2+n*π)

Final Answer

y=sin(ln(x))

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