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  5. Logarithms - Transformations

Logarithms - Transformations

There are various transformations that can be applied to a parent logarithmic function in order to change its core attributes. A transformed function can be expressed algebraically as:

\(f(x) = a\log[k(x - d)] + c\)

  • \(a\) represents the vertical stretch/compression factor
  • \(k\) represents the horizontal stretch/compression factor
  • \(d\) represents the horizontal shift. This also affects the Horizontal Asymptote
  • \(c\) represents the vertical shift

Vertical Stretch/Compression

The value of \(a\) will determine whether the transformed function will be vertically stretched or compressed:

  • If \(a > 1\), the function will be vertically stretched by a factor of \(a\)
  • If \(0 < a < 1\), the function will be compressed by a factor of \(a\)

\(a > 0\)

Parent Logarithmic function graphed against vertically stretched Logarithmic function.

\(0 < a < 1\)

Parent Logarithmic function graphed against vertically compressed Logarithmic function.

Horizontal Stretch/Compression

The value of \(k\) will determine whether the transformed function will be horizontally stretched or compressed. This also affects the location of the function's Horizontal Asymptote:

  • If \(0 < k < 1\), the function will be horizontally stretched by a factor of \(1/k\)
  • If \(k > 1\), the function will be horizontally compressed by a factor of \(1/k\)
  • The period is related to the \(k\) value through the equation \(p = 2\pi/l\)

\(0 < k < 1\)

Parent Logarithmic function graphed against horizontally stretched Logarithmic function.

\(k > 1\)

Parent Logarithmic function graphed against horizontally compressed Logarithmic function.

Horizontal Shift

The value of \(d\) will determine whether the transformed function will be shifted left or right:

  • If \(d > 0\), the function will be shifted right
  • If \(d < 0\), the function will be shifted left

\(d > 0\)

Parent Logarithmic function graphed against Logarithmic function shifted to the right.

\(d < 0\)

Parent Logarithmic function graphed against Logarithmic function shifted to the left.

NOTE: The horizontal shift will affect the function's Vertical Asymptote.


Vertical Shift

The value of \(c\) will determine whether the transformed function will be shifted upward or downward:

  • If \(c > 0\), the function will be shifted upward
  • If \(c < 0\), the function will be shifted downward

\(c > 0\)

Parent Logarithmic function graphed against Logarithmic function shifted upwards.

\(c < 0\)

Parent Logarithmic function graphed against Logarithmic function shifted downwards.

Reflections

  • If \(a < 0\), the function will either be vertically stretched or compressed with a reflection in the \(x\)-axis
  • If \(k < 0\), the function will either be horizontally stretched or compressed with a reflection in the \(y\)-axis

\(a < 0\)

Parent Logarithmic function graphed against Logarithmic function reflected in the x-axis.

\(k < 0\)

Parent Logarithmic function graphed against Logarithmic function reflected in the y-axis.

Example

For the function \(f(x) = 2\log(x - 5) + 3\):

  1. Identify the transformations that were applied
  2. Determine the Horizontal Asymptote
  3. Determine the domain and range
  4. Sketch the transformed function

i. We can now determine the transformations as such:

  • \(\boldsymbol{a = 2}\); vertical stretch by a factor of \(2\)
  • \(\boldsymbol{d = 5}\); shifted right by \(5\) units
  • \(\boldsymbol{c = 3}\); shifted upward \(3\) units

ii. Since \(d = 5\), we can determine the Horizontal Asymptote is located at \(\boldsymbol{x = 5}\).

iii. We can identify the domain and range as such:

  • Domain: \(\{x\in\mathbb{R} | x > 5\}\)
  • Range: \(\{y\in\mathbb{R}\}\)

iv. We can draw our transformed graph as such:

Graph of transformed Logarithmic function expressed as f(x)=2log(x-5)+3.

For the function \(g(x) = -\log(0.5x + 1.25) - 4\):

  1. Identify the transformations that were applied
  2. Determine the Horizontal Asymptote
  3. Determine the domain and range
  4. Sketch the transformed function

i. After shifting some of the values in the transformed equation, we can represent it as:

\(g(x) = -\log(0.5(x + 2.5) - 3\)

  • \(\boldsymbol{a = -1}\); reflection in the \(x\)-axis
  • \(\boldsymbol{k = 2}\); horizontal stretch by a factor of \(2\)
  • \(\boldsymbol{d = -2.5}\); shifted left by \(2.5\) units
  • \(\boldsymbol{c = -3}\); shifted downward by \(3\) units

ii. Since \(d = -2.5\), we can determine the Horizontal Asymptote is located at \(\boldsymbol{x = -2.5}\).

iii. We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R} | x > -2.5\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R}\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed Logarithmic function expressed as g(x)=-log(0.5(x+2.5))-3.
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