Question

Sketch a graph of $f$ $$f(x)=\log _{2} x$$

   Sketch a graph of $f$
$$f(x)=\log _{2} x$$

College Algebra with Modeling and Visualization

College Algebra with Modeling and Visualization

Gary K. Rockswold 4th Edition

Chapter 5, Problem 63

Instant Answer

Solved by Expert Steven Clarke

10/15/2021

Step 1

The graph of a logarithmic function is a smooth curve that starts from the bottom left, gets steeper as it moves to the right, and never touches the y-axis. This is because the logarithm of a number approaches negative infinity as the number approaches zero. Show more…

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Sketch a graph of $f$ $$f(x)=\log _{2} x$$

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Key Concepts

x-Intercept

A key feature of logarithmic functions is that they cross the x-axis at x = 1, since the logarithm of 1 is 0 for any positive base. This intercept serves as a reference point when sketching the graph.

Asymptotic Behavior

The graph of a logarithmic function has a vertical asymptote at x = 0, meaning that as x approaches 0 from the right, the function's value decreases without bound. This characteristic is crucial for depicting the graph correctly.

Graph Sketching Techniques

To sketch the graph of a logarithmic function, plot the characteristic point (1, 0), draw the vertical asymptote at x = 0, and indicate the increasing or decreasing trend based on the base. Additional points can be calculated to depict the gradual increase, ensuring accurate representation of the logarithm’s shape.

Logarithmic Function

A logarithmic function is the inverse of an exponential function and represents the power to which a base number must be raised to produce a given number. It grows slowly for larger input values and decreases rapidly near zero, and its behavior is largely determined by its base.

Domain and Range

For a logarithmic function, the domain is all positive real numbers (x > 0) because logarithms are only defined for positive arguments, while the range spans all real numbers, reflecting that the output can be any real number.

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