Question

Sketch the graph of the function and state its domain. $$ f(x)=3 \ln x $$

Short answer

The domain of the function \(f(x)=3 \ln x\) is \(x>0\). The graph is a vertical stretch by a factor of 3 of the standard log graph, intersecting the x-axis at x=1 and increasing indefinitely as x increases, but it never crosses the y-axis.

Step-by-step solution

State the Domain

The domain of a log function is always x > 0. Therefore the domain for this function is \(x > 0\).

Identify the Key Characteristics

As per the general rules of transformations, a value multiplied with the standard function stretches or compresses the graph vertically. Therefore, the function \(f(x) = 3 \ln x\) will be a vertically stretched version of the standard logarithm function \(\ln x\), by a factor of 3.

Sketch the Graph

Start by sketching the standard logarithm function \(\ln x\), which starts from the point (1,0) and increases gradually, the line getting steeper as x increases. Then stretch the graph vertically by a factor of 3, meaning every y-coordinate is tripled. The graph intersects the x-axis at x=1 and continues to increase indefinitely as x increases, but it never crosses the y-axis.