Characteristics of Log Functions

​Characteristics of Logarithmic Functions

​Algebra 2

Logs are the inverse of Exponential Functions--their graphs are inverted

2^x is in red

log₂​ x is in blue

Notice the asymptotes?​

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​What else has switched?

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Graphs of Log Functions

We will focus on what a, h, and k do to affect the parent function

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​Transformations of Logs

The parent graph, log₂​(x)

The transformed graph,

log₂​(x+3)

Which way did the graph move? Where is the asymptote?​

​what is the h-value for the red graph?

The asymptote, h

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Multiple Choice

Identify the asymptote of the transformed graph $$\log_2\left(x+3\right)$$  shown in red in the graph

• $$x=-3$$  

• $$x=3$$  

• $$y=3$$  

• $$y=-3$$  

Multiple Choice

Which of the equations could best describe the graph?

• $$\log_2\left(x+1\right)$$  

• $$\log_2\left(x-1\right)$$  

• $$\log_2\left(x\right)+1$$  

• $$\log_2\left(x\right)-1$$  

Match

Match the following log functions to their horizontal shift (which way will the asymptote shift)

• $$\ln\left(x\right)$$  

• $$\ln\left(x-4\right)$$  

• $$\ln\left(x+5\right)$$  

No Horizontal Shift

Shift Right

Shift Left

• No Horizontal Shift

• Shift Right

• Shift Left

Multiple Choice

Identify the asymptote for the function ℎ(𝑥) = 𝑙𝑛(𝑥 − 1) + 3.

• $$x=3$$  

• $$x=-1$$  

• $$y=1$$  

• $$x=1$$  

Multiple Choice

Which graph correctly shows the asymptote of $$f(x)=\log(x+2)+3$$  

Multiple Choice

Which function below has a different asymptote than the rest of the functions?

• $$a\left(x\right)$$  

• $$b\left(x\right)$$  

• $$c\left(x\right)$$  

• $$d\left(x\right)$$  

Domain: x > -2

​h will affect the domain of log functions

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​Remember: can't take the log of ZERO or NEGATIVE arguments

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The vertical asymptote is either the left or right boundary of domain​​

Match

Match the following graphs with their domains

Domian: x > -2

Domian: x > 3

Domian: x > 0

Domian: x > 2

• Domian: x > -2

• Domian: x > 3

• Domian: x > 0

• Domian: x > 2

Multiple Choice

Identify the domain of the graph

• Domain: $$\left(asymptote,\ \infty\right)$$  

• Domain: $$\left(-\infty,\ asymptote\right)$$  

• Domain: $$\left[asymptote,\ \infty\right]$$  

• Domain: $$\left[-\infty,\ asymptote\right]$$  

Multiple Choice

Of the functions shown, which is the ONLY possible function to describe the graph shown?

• $$-\ln\left(x+3\right)$$  

• $$-\ln\left(x-3\right)$$  

• $$-\ln\left(x\right)-3$$  

• $$-\ln\left(x\right)+3$$  

Multiple Choice

Which of the following has a domain of all real numbers, x > 2 ?

• y = log₂ x

• y = log₃ (x+2)

• y = log₄ (x - 2)

• y = log₇ x - 2

Multiple Choice

Which of the following is the sketch of the graph of y = -log₃ (x+5) ?

Multiple Choice

What is the domain of $$y=\log_3\left(x-3\right)-2$$  

• $$\left(-\infty,\infty\right)$$  

• $$\left(-2,\infty\right)$$  

• $$\left(3,\infty\right)$$  

• $$\left(-3,\infty\right)$$  

Multiple Choice

What is the range of $$y=\log_3\left(x-3\right)-2$$  

• $$\left(-\infty,\infty\right)$$  

• $$\left(2,\infty\right)$$  

• $$\left(-2,\infty\right)$$  

• $$\left(-\infty,2\right)$$  

Multiple Choice

What is the domain?

• (-∞, ∞)
• (0, ∞)
• [1, ∞)
• (1, ∞)

Multiple Choice

What is the range?

• (-∞, ∞)
• (0, ∞)
• [1, ∞)
• (1, ∞)

Multiple Choice

Graph $$\ln(x+1)$$