Sketch graphs of: (a) f(x)=log3(x+2); (b) F(x)=3-log2 x; (c) g(x)=ln|x|; (d) G(x)=-ln(-x);

Name: Problem 22, Chapter 18: Schaum's outlines: precalculus
Uploaded: 2022-04-05T16:22:58-08:00
Duration: 1 min 4 s
Channel: Carson Merrill
Description: Sketch graphs of: (a) f(x)=log3(x+2); (b) F(x)=3-log2 x; (c) g(x)=ln|x|; (d) G(x)=-ln(-x);

step 1 Our goal for this problem is to graph the logarithmic function f of x equals log base 3 of x. Th

Sketch graphs of: (a) f(x)=log3(x+2); (b) F(x)=3-log2 x; (c)...

Key Concepts

Absolute Value Transformations

When the absolute value is applied to the input of a logarithmic function, it reflects the part of the graph corresponding to negative values of the variable into the positive side. This transformation broadens the domain of the logarithmic function by allowing both positive and negative inputs, except where the absolute value causes the argument to be zero.

Function Transformations

Function transformations involve shifting, reflecting, stretching, or compressing the graph of a basic function. For logarithmic functions, horizontal shifts are applied when the input variable is adjusted (such as adding or subtracting a constant), while vertical shifts occur when a constant is added or subtracted from the entire function. These transformations enable the graph to move to different positions on the coordinate plane.

Reflection Transformations

Reflection transformations in logarithmic functions occur when there is a negation applied either to the function as a whole or to its input. Reflecting the graph vertically (by multiplying the function by -1) or horizontally (by negating the input) alters the orientation of the curve. Recognizing these reflections is important for determining the behavior of the function and for accurate graphing.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined only for positive arguments. They have a characteristic curve that increases slowly for large input values and decreases steeply as the input approaches zero from the right. Understanding the basic shape and behavior of logarithmic functions is essential in graphing and analyzing these functions.

Domain of Logarithmic Functions

The domain of a logarithmic function is determined by the condition that the input to the logarithm must be positive. When the argument of the logarithm is transformed (for example, shifted or includes an absolute value), it is important to adjust the domain accordingly to ensure that the expression inside the log remains greater than zero.

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