Graph natural log of 3x

Problem

y=ln(3*x)

Solution

1. Identify the domain of the function. Since the argument of a natural logarithm must be positive, we set 3*x>0 which means x>0

2. Determine the vertical asymptote by finding where the argument equals zero. As x approaches 0 from the right, y approaches −∞ so the vertical asymptote is x=0

3. Find the x-intercept by setting y=0 Solving 0=ln(3*x) gives e0=3*x which simplifies to 1 = 3x,o*r = \frac{1}{3}.T*h*e*i*n*t*e*r*c*e*p*t*i*s()\frac{1}{3}, 0)$.

4. Calculate additional points to determine the shape. When x=1 y=ln(3)≈1.10 When x=e/3≈0.91 y=ln(e)=1

5. Analyze the behavior as x increases. The function is strictly increasing and concave down because the first derivative d(y)/d(x)=1/x is positive and the second derivative d2(y)/(d(x)2)=−1/(x2) is negative for all x>0

Final Answer

y=ln(3*x)

Want more problems? Check here!