Graph natural log of natural log of x

Problem

y=ln(ln(x))

Solution

1. Identify the domain of the function. For the outer natural log to be defined, the inner expression ln(x) must be greater than zero.

ln(x)>0

x>e0

x>1

2. Determine the x-intercept by setting y=0

ln(ln(x))=0

ln(x)=e0

ln(x)=1

x=e

3. Analyze the vertical asymptote as x approaches the boundary of the domain from the right.

(lim_x→1)(ln(ln(x)))=−∞

Vertical asymptote at *x=1$.4.∗∗D*e*t*e*r*m*i*n*e*t*h*e*e*n*d(b)*e*h*a*v*i*o*r∗∗a*s$x$i*n*c*r*e*a*s(e)*s(t)*o*w*a*r*d(i)*n*ƒ*i*n*i*t*y.$

\lim_{x \to \infty} \ln(\ln(x)) = \infty

T*h*e*ƒ*u*n*c*t*i*o*n*g*r*o*w*s(v)*e*r*y*s(l)*o*w*l*y*b*u*t*w*i*t*h*o*u*t*b*o*u*n*d.5.∗∗F*i*n*d(t)*h*e*d(e)*r*i*v*a*t*i*v*e∗∗t*o*d(e)*t*e*r*m*i*n*e*t*h*e*s(l)*o*p*e*a*n*d(c)*o*n*c*a*v*i*t*y.

\frac{d \ln(\ln(x))}{dx} = \frac{1}{x \ln(x)}

Since x > 1 and ln(x) > 0 for the entire domain, the function is always increasing. 6. **Sketch the graph** starting from the vertical asymptote at x = 1, passing through the point (e, 0), and continuing to increase slowly with a concave down shape.

Final Answer

y = ln(ln(x)) for x > 1

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