Graph y = log base 3 of x

Problem

y=(log_3)(x)

Solution

1. Identify the domain, which is the set of all possible x values. Since the argument of a logarithm must be positive, the domain is x>0 This implies a vertical asymptote at x=0

2. Determine the x-intercept by setting y=0 Solving 0=(log_3)(x) gives 3=x so the intercept is at (1,0)

3. Calculate key points to determine the shape of the curve. Choose values of x that are powers of the base 3

• If x=1/3 then y=(log_3)(1/3)=−1 Point: (1/3,−1)

• If x=3 then y=(log_3)(3)=1 Point: (3,1)

• If x=9 then y=(log_3)(9)=2 Point: (9,2)

4. Analyze the behavior of the function. As x approaches 0 from the right, y approaches −∞ As x increases, y increases slowly.

5. Sketch the graph by plotting the points (1,0) (3,1) and (1/3,−1) then drawing a smooth curve that approaches the yaxis as a vertical asymptote.

Final Answer

y=(log_3)(x)

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