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Wize High School Grade 12 Pre-Calculus Textbook > Logarithmic Functions

Transformations of Logarithmic Functions

Characteristics of Logarithmic FunctionsPrevious Section
Applications of Logarithmic FunctionsNext Section
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Transformations of Logarithmic Functions


Let y=alog⁡b(c(x−h))+ky=a\log_{b}(c(x-h))+k be the transformed function of y=f(x), y=f(x),~ where a, b, c, h, k are real numbers. Then:

HorizontalVerticalc<0‾:a<0‾:horizontal reflection about thevertical reflection about they-axisx-axis∣c∣>1‾:∣a∣>1‾:horizontal compression of  1c  unitsvertical expansion of ’a’ units∣c∣<1‾:∣a∣<1‾:horizontal expansion of  1c  unitsvertical compression of ’a’ unitsh>0‾:k>0‾:horizontal translation ’h’ units rightvertical translation ’k units uph<0‾:k<0‾:horizontal translation ’h’ units leftvertical translation ’k’ units down\begin{array}{l c c l} \text{Horizontal}&&&\text{Vertical}\\\\ \underline{c<0}:&&&\underline{a<0}:\\ \text{horizontal reflection about the}&&&\text{vertical reflection about the}\\ \text{y-axis}&&&\text{x-axis}\\\\\\ \underline{|c|>1}:&&&\underline{|a|>1}:\\ \text{horizontal compression of}~~\displaystyle\frac{1}{c}~~\text{units}&&&\text{vertical expansion of 'a' units}\\\\\\ \underline{|c|<1}:&&&\underline{|a|<1}:\\ \text{horizontal expansion of}~~\displaystyle\frac{1}{c}~~\text{units}&&&\text{vertical compression of 'a' units}\\\\\\ \underline{h>0}:&&&\underline{k>0}:\\ \text{horizontal translation 'h' units right}&&&\text{vertical translation 'k units up}\\\\\\ \underline{h<0}:&&&\underline{k<0}:\\ \text{horizontal translation 'h' units left}&&&\text{vertical translation 'k' units down} \end{array}

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Example

The graph of y=−2log⁡2(12(x−1))+1y=-2\log_{2}{\big(\frac{1}{2}(x-1)\big)}+1 has the following transformations:
  • Horizontal:
  • Expansion by a factor of 2
  • Translation 1 unit right
  • Vertical:
  • Reflection about the x-axis
  • Expansion by a factor of 2
  • Translation 1 unit up

Parent Function: y=log⁡2xy=\log_{2}x
Table of values for parent function:

xy1/4−21/2−1102142\begin{array}{|c|c|}\hline x&y\\\hline 1/4&-2\\\hline 1/2&-1\\\hline 1&0\\\hline 2&1\\\hline 4&2\\\hline \end{array}

Transformed Function: y=−2log⁡2(12(x−1))+1y=-2\log_{2}{\big(\frac{1}{2}(x-1)\big)}+1
Table of values for transformed function:

xy1.5523315−19−3\begin{array}{|c|c|}\hline x&y\\\hline 1.5&5\\\hline 2&3\\\hline 3&1\\\hline 5&-1\\\hline 9&-3\\\hline \end{array}

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The vertical asymptote is x=1x=1
The domain is x>1x>1
The range is y∈Ry\in\mathbb{R}

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Example: Transformations of Logarithmic Functions

Sketch a graph of the function y=−2log⁡(2x−4)+1y=-2\log(2x-4)+1, stating the transformations, any asymptotes, the domain, and the range.

The function can be rewritten as y=−2log⁡(2(x−2))+1y=-2\log(2(x-2))+1.


The Transformations

Horizontal
  • Compression by a factor of 12\dfrac{1}{2}
  • Translation 2 units right

Vertical
  • Reflection about the x-axis
  • Expansion by a factor of 2
  • Translation 1 unit up

Table of Values for the Parent Function y=log⁡x\colorThree{y=\log{x}}

xy1/100−21/10−110101100210003\begin{array}{c|c} x&y\\\hline\\ 1/100&-2\\\\ 1/10&-1\\\\ 1&0\\\\ 10&1\\\\ 100&2\\\\ 1000&3 \end{array}


Table of Values & Graph for the Transformed Function

xy2.00552.0535/217−152−3502−7\begin{array}{c|c} x&y\\\hline\\ 2.005&5\\\\ 2.05&3\\\\ 5/2&1\\\\ 7&-1\\\\ 52&-3\\\\ 502&-7 \end{array}


Vertical Asymptote: x=2\colorFive{x=2}
Domain: x>2\colorFive{x>2}
Range: y∈R\colorFive{y\in\mathbb{R}}

Practice: Transformations of Logarithmic Functions

The function y=log⁡xy=\log_{}{x} is horizontally reflected about the y-axis, translated 1 unit left, and vertically compressed by a factor of 12\dfrac{1}{2}.
What's the equation of the transformed function?

Practice: Transformations of Logarithmic Functions

Which of the following is a graph of y=3log⁡(−x+3)−1y=3\log(-x+3)-1?

Practice: Transformations of Logarithmic Functions

If the point (−3,5)(-3,5) is on the function y=2log⁡3(3−2x)+1y=2\log_{3}(3-2x)+1, then what point must be on y=3xy=3^x?
Extra Practice
Transformations of Logarithmic Functions
Match the function with the correct graph.
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
If the point (a,b)(a,b) is on the function y=log⁡5(4−3x)−2y=\log_5(4-3x)-2, then what point must be on y=5xy=5^x?
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
If the point (1,1)(1,1) is on the function y=−3log⁡2(x+1)−2y=-3\log_{2}(x+1)-2, then what point must be on y=2xy=2^x?
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
Is (−3,−2)(-3,-2) a graph of y=−log⁡2(−x+5)+1y=-\log_2{(-x+5)}+1?
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
Is (3,0)(3,0) a point on the graph of y=−log⁡(x−2)+3y=-\log(x-2)+3?
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
The function y=log⁡xy=\log_{}{x} is vertically expanded by a factor of 2, horizontally compressed by a factor fo 34\dfrac{3}{4}, and translated 5 up.
What's the equation of the transformed function?
Transformations of Logarithmic Functions
Practice: Transformations of Logarithmic Functions
The function y=log⁡xy=\log_{}{x} is vertically reflected about the x-axis, translated 3 units right, and horizontally expanded by a factor of 44.
What's the equation of the transformed function?