Graph natural log of 6x

Problem

y=ln(6*x)

Solution

1. Identify the domain of the function. Since the argument of a natural logarithm must be positive, we set 6*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(6*x) gives e0=6*x which simplifies to 1 = 6x,s(o)*t*h*e*i*n*t*e*r*c*e*p*t*i*s(a)*t = \frac{1}{6}$.

4. Calculate additional points to determine the shape. For example, when x=e/6 y=ln(e)=1 When x=1 y=ln(6)≈1.79

5. Sketch the curve starting near the vertical asymptote x=0 passing through (1/6,0) and increasing slowly as x increases.

Final Answer

y=ln(6*x)

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