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## Drawing log graphs

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

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 $y$

$y=log_{10}x+2$

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.

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

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.

Questions