Question

sketch the graph of the following function, and state its domain and range. f(x) = log₈(x)

Explanation:

Step1: Recall Logarithmic Function Properties

For a logarithmic function \( f(x) = \log_b(x) \), the domain is determined by the argument \( x > 0 \) (since we can't take the logarithm of a non - positive number). The range of a logarithmic function \( \log_b(x) \) (where \( b>0, b
eq1 \)) is all real numbers, \( (-\infty, \infty) \), because as \( x \) approaches \( 0^+ \), \( \log_b(x) \) approaches \( -\infty \), and as \( x \) approaches \( \infty \), \( \log_b(x) \) approaches \( \infty \), and it takes on all values in between. For the graph of \( y=\log_8(x) \), since the base \( b = 8>1 \), the function is increasing. The vertical asymptote is the line \( x = 0 \) (the y - axis), and the function passes through the point \( (1,0) \) because \( \log_b(1)=0 \) for any valid base \( b \), and also through the point \( (8,1) \) because \( \log_8(8) = 1 \).

Step2: Determine Domain

For \( f(x)=\log_8(x) \), the argument of the logarithm must be positive. So, the domain is the set of all real numbers \( x \) such that \( x>0 \), which in interval notation is \( (0,\infty) \).

Step3: Determine Range

As \( x \) takes on all positive real values, \( \log_8(x) \) can take on any real number value. For example, if we want to find a value \( y \) such that \( \log_8(x)=y \), we can rewrite it in exponential form as \( x = 8^y \). Since \( y \) can be any real number (and for any real \( y \), \( 8^y>0 \)), \( x \) will be a positive real number, meaning that \( y=\log_8(x) \) can take on any real number value. So the range is \( (-\infty,\infty) \).

Step4: Sketch the Graph (Summary)

• Vertical Asymptote: Draw a dashed vertical line at \( x = 0 \) (the y - axis).
• Key Points: Plot the point \( (1,0) \) (since \( \log_8(1)=0 \)) and \( (8,1) \) (since \( \log_8(8) = 1 \)).
• Shape: Since the base \( 8>1 \), the graph is an increasing curve that approaches the vertical asymptote \( x = 0 \) as \( x \) approaches \( 0^+ \) and increases without bound as \( x \) approaches \( \infty \).

Answer:

• Domain: \( (0,\infty) \)
• Range: \( (-\infty,\infty) \)
• Graph: An increasing logarithmic curve with a vertical asymptote at \( x = 0 \), passing through \( (1,0) \) and \( (8,1) \).