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Drawing log graphs

First lets consider the basic log graph y=log_{10} x

Log graph features

The important features

  • The graph has an asymptote of x=0
  • It intersects the x axis at x=1, because the log of 1 is zero
  • It passes through the point (10,1), because 10^1=1.  Notice 10 is the base
  • The curve is always increasing, but at a slower and slower rate

Conclusions

The asymptote will always be x=0 unless the graph is translated \Rightarrow y=log_{10}(x+a) has an asymptote at x=-a

y=1 when x equals the base number unless the graph has been translated \Rightarrow y=log_a(x) goes through (a,1)

The graph goes through (1,0) unless translated \Rightarrow y=log_{10}(x+a)+b goes through the point (10-a,1+b)

Examples

For each of the function below give the asymptote, the coordinates where the point (1,0) has moved to and the coordinates of where the point (base,1) has moved to.

y=log_5x

The asymptote has not changed since we have not added to x so the asymptote is x=0

The point related to the base is (5,1)

The point (1,0) has not moved since we have not added to x or y

Confirms claims above

y=log_{10}x+2

The asymptote has not changed since we have not added to x so the asymptote is x=0.

The point related to the base is (10,3) because we have added 2 to the function so the graph have moved up 2.

The point (1,0) has moved to (1,2) since we have added 2 to the function so the graph have moved up 2.

Confirms points made above

y=3log_{10}(x-1)+2

The asymptote is x=1 because we have subtracted 1 from x and this shifts the graph 1 place to the right.

The point related to the base is (11,5) because the y values are 3 times bigger and then we add 2.

The point (1,0) has moved to (1,2) since we need to multiply y by 3 (no effect here) and then add 2.

Confirms points made above

Questions

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