Graphs of Logarithmic Functions

One more fun day in Section 3.3b

Let’s start with Analysis of the Natural Logarithmic Function:Let’s start with Analysis of the Natural Logarithmic Function:

The graph: lnf x x

Domain: 0, Range: , Continuous on 0,Increasing on 0,

No Symmetry Unbounded

No Local Extrema No Horizontal Asymptotes

Vertical Asymptote: 0x

End Behavior: lim lnx

x

The “Do Now” The “Do Now” Analysis of the Natural Logarithmic Function Analysis of the Natural Logarithmic Function

The graph: lnf x xNote: Any other logarithmicNote: Any other logarithmicfunctionfunction logbg x xwith with bb > 1 has the same domain, > 1 has the same domain,range, continuity, inc. behavior,range, continuity, inc. behavior,lack of symmetry, and otherlack of symmetry, and othergeneral behavior of the naturalgeneral behavior of the naturallogarithmic function!!!logarithmic function!!!

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

1. ln 2g x x Trans. left 2 Trans. left 2 The graph? The graph?

2. ln 3h x x Reflect across the Reflect across the yy-axis,-axis,Trans. right 3 Trans. right 3 The graph? The graph?

ln 3x

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

3. 3logg x xVert. stretch by 3 Vert. stretch by 3 The graph? The graph?

4. 1 logh x x Trans. up 1 Trans. up 1 The graph? The graph?

Describe how to transform the graph of y = ln(x) or y = log(x)into the graph of the given function. Sketch the graph by handand support your answer with a grapher.

5. 2ln 2 2 3g x x

Trans. right 1, Horizon. shrink by 1/2,Trans. right 1, Horizon. shrink by 1/2,Reflect across both axes, Vert. stretch by 2,Reflect across both axes, Vert. stretch by 2,Trans. up 3 Trans. up 3 The graph??? The graph???

2ln 2 1 3x

Graph the given function, then analyze it for domain, range,continuity, increasing or decreasing behavior, boundedness,extrema, symmetry, asymptotes, and end behavior.

1. log 2f x x D: 2, R: ,

Continuous Dec: 2, No Symmetry Unbounded No Local Extrema

Asy: 2x E.B.: limx

f x

Graph the given function, then analyze it for domain, range,continuity, increasing or decreasing behavior, boundedness,extrema, symmetry, asymptotes, and end behavior.

2. 5ln 2 3f x x

D: , 2 R: ,

Continuous Dec: , 2 No Symmetry Unbounded No Local Extrema

Asy: 2x E.B.: limx

f x