Exer. 35-46: Sketch the graph of f f(x)=log3(3 x)

Name: Problem 35, Chapter 4: Precalculus: Functions and Graphs
Uploaded: 2020-10-22T16:27:19-08:00
Duration: 4 min 36 s
Channel: Raushan Kumar
Description: Exer. 35-46: Sketch the graph of f f(x)=log3(3 x)

Key Concepts

Logarithmic Function Properties

Logarithmic functions are the inverses of exponential functions and are defined only for positive arguments. They have a vertical asymptote along the y-axis (x=0) and extend indefinitely in the y-direction. The graph typically passes through the point (1, 0) when expressed in the form log_b(x), showcasing these inherent characteristics.

Logarithm Rules and Properties

Logarithmic rules, such as the product rule (log_b(ax) = log_b(a) + log_b(x)), allow us to decompose complex logarithmic expressions into simpler components. This decomposition is useful for identifying shifts and other modifications in the graph of the function.

Function Transformations

Function transformations involve modifying the basic graph by vertical or horizontal shifts, stretches, and reflections. In the case of logarithmic functions, rewriting the function to include an added constant (for instance, as log_b(x) + c) indicates a vertical translation, which shifts the entire graph up or down without changing its overall shape, asymptote, or domain.

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