Graph y = log base 8 of x

Problem

y=(log_8)(x)

Solution

1. Identify the parent function as a logarithmic function with base b=8 Since b>1 the graph is increasing and passes through the point (1,0)

2. Determine the domain and range. The argument of the logarithm must be positive, so the domain is x>0 The range of any logarithmic function is all real numbers, (−∞,∞)

3. Find the vertical asymptote. As x approaches 0 from the right, y approaches −∞ meaning there is a vertical asymptote at x=0

4. Calculate key points to plot. Choose values of x that are powers of 8 to find integer y values:

• If x=1/8 then y=(log_8)(1/8)=−1 Point: (1/8,−1)

• If x=1 then y=(log_8)(1)=0 Point: (1,0)

• If x=8 then y=(log_8)(8)=1 Point: (8,1)

• If x=64 then y=(log_8)(64)=2 Point: (64,2)

5. Sketch the curve by starting near the vertical asymptote x=0 in the fourth quadrant, passing through the identified points, and continuing to rise slowly as x increases.

Final Answer

y=(log_8)(x)

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