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Exponential and Logarithmic Functions

Basic Logarithmic Function & Its Characteristics

The following are graphs of the basic logarithmic function, \(f(x)=log_b(x), \\ b>0, \\  b\ne 1\):

\(f(x)=log_b(x)\) where \(0<b<1\)

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\(f(x)=log_b(x)\) where \(b>1\)

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Characteristics of the Logarithm Function

Using the above graphs, we can identify some characteristics of the basic logarithmic function \(f(x)=log_b(x), b>0, b \ne 1\), including:

  • Domain: \(D:\{x \in \mathbb{R} \ | \ x > 0\}\)
  • Range: \(R:\{y \in \mathbb{R}\}\) 
  • Vertical asymptote: \(x=0\)
  • \(x\)-intercept: \(1\)

Tip: If you recall the characteristics of the basic exponential function (\(f(x)=b^x\)) graph (which can be found here), you'll see that the the basic exponential and logarithmic functions are very similar, and are, in fact, related. This is because both functions are inverses of each other, so their characteristics are also the inverse of each other.

So, if you can recall the characteristics of one function, you can then use the fact that they're inverse functions to recall the other functions' characteristics. Specifically, characteristics associated with the \(x\) and \(y\) variables are switched.

The related characteristics are summarized in the following tables:

                                           

Graphing Logarithmic Functions with Transformations

Given the graph of the base/parent function \(f(x)=log_b(x)\), we are able to graph any logarithmic function of the form:

\(f(x)=alog_b(k(x-d))+c\), for any \(a,k,c,d\in \mathbb{R}\) 

by applying transformations to the parent graph. Each of the four parameters, \(a,b,c,d\) correspond to certain transformations, as summarized in the following table:

Function Notation/Parameter Corresponding Transformation(s) Coordinate Point Transformation

\(f(x)=\)\(\ a\)\(log_b(k(x-d))+c\)

Vertical stretch/compression & reflection

  • If \(a<0\), vertical reflection (i.e. over the x-axis)
  • If \(|a|>1\), vertical stretch by a factor of \(|a|\)
  • If \(0<|a|<1\), vertical compression by a factor of \(|a|\)

\((x,y) \rightarrow\) \((x,\)\(\ a\)\(y)\)

\(f(x)=alog_b(\)\(k\)\((x-d))+c\)

Horizontal stretch/compression & reflection

  • If \(k<0\), horizontal reflection (i.e. over the y-axis)
  • If \(|k|>1\), horizontal compression by a factor of \(|\frac{1}{k}|\)
  • If \(0<|k|<1\), horizontal stretch by a factor of \(|\frac{1}{k}|\)

\((x,y) \rightarrow(\)\(\frac{1}{k}\)\(x,y)\)

\(f(x)=alog_b(k(x\ - \)\(\ d\)\())+c\)

Horizontal shift/translation left or right by \(d\) units

  • If \(d>0\), shift right by \(d\) units
  • If \(d<0\), shift left by \(d\) units

\((x,y) \rightarrow\)\(\ (x\)\(+ d\)\(,y)\)

\(f(x)=alog_b(k(x-d))\ +\)\(\ c\)

Vertical shift/translation up or down by \(c\) units

  • If \(c>0\), shift up by \(c\) units
  • If \(c<0\), shift down by \(c\) units

\((x,y) \rightarrow\) \((x,y\)\(+ c\)\()\)

Tip: To help remember what parameters correspond to which transformations, note that parameters "outside" of the function \(f(x)=log_b(x)\) (i.e. \(a, c\)) transform the \(y\) or vertical value, while the ones "inside" the function (i.e. \(k, d\)) transform the \(x\) or horizontal value.

Notes:

  • After transforming, the vertical asymptote is located at \(x=d\)  since the value we apply the logarithm to cannot equal zero. (Why?)
  • The domain of the transformed function is dependent on whether it lies to the left or to the right of the asymptote:
    • If the function is on the left of the asymptote, the domain is \(x<d\)
    • If the function is on the right of the asymptote, the domain is \(x>d\)
  • Regardless of transformations, the range of a logarithmic function will always be \(\{y\in\mathbb{R}\}\) (Why?)

Examples

Example: Sketch the function \(f(x)=2log_3(x-1)-2\) by first:
a) Graphing the parent function, then applying transformations b) Plotting any key points and drawing the asymptote.

Solutions:

a) Graphing the parent function, then applying transformations.

1. In this case, the parent function is \(f(x)=log_3(x)\), which has a vertical asymptote of \(x=0\). So we first sketch this function:

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*Sketch this graph by creating a table values then plotting the points, or by finding the x-intercept, vertical asymptote and another point on the graph, then sketching the function.

2. Identifying the transformation parameters, we see that we have \(a=2\), so we have a vertical stretch by a factor of 2:

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3. Next, we have \(d=1\), so we have a horizontal shift to the right by 1 unit. This also shifts our original vertical asymptote right to \(x=1\):

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4. Finally, we have \(c=-2\), so we have a vertical shift down by 2 units:

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*Note: It is also possible to graph this function by transforming each individual point from the parent function or by applying all the transformations graphically at the same time (instead of one-by-one like we did above).


We compare the graphs of the two functions, \(f(x)=log_3(x)\) and \(f(x)=2log_3(x-1)-2\), and summarize their characteristics in the following table:

\(f(x)=log_3(x)\) \(f(x)=2log_3(x-1)-2\)
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Domain: \(D:\{x\in\mathbb{R}\ | \ x>0\}\)

Range: \(R:\{y\in\mathbb{R}\}\)

Vertical asymptote: \(x=0\)

\(x\)-intercept: \((1,0)\)

\(y\)-intercept: None

Domain: \(D:\{x\in\mathbb{R}\ | x>1\}\)

Range: \(R:\{y\in\mathbb{R}\}\)

Vertical asymptote: \(x=1\)

\(x\)-intercept: \((4,0)\)

\(y\)-intercept: None

b) Plotting any key points and drawing the asymptote:

1. The function \(f(x)=2log_3(x-1)-2\) has:

  • \(x\)-intercept at \((4,0)\),
  • No \(y\)-intercept
  • points at \((2,-2)\) and \((10,2)\), and
  • vertical asymptote \(x=1\).

We can plot and draw these characteristics on the Cartesian plane first:

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2. After plotting/drawing these characteristics, we can draw a smooth curve through the points and approaching the asymptote, in order to finish sketching the function:

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