# Swing the Logarithmic Curve around (1, 0)

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The logarithmic function to the base , where and , is defined by if and only if ; the domain is and the range is .

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Move the slider; the base of the logarithm changes and you see its graph swing around the point .

Closely observe the two cases and . Also notice where the blue curve lies in relation to the common logarithm (base 10) and the natural logarithm .

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Contributed by: Abraham Gadalla (March 2011)
Open content licensed under CC BY-NC-SA

## Details

When considering the common logarithm (i.e., base 10), we notice that as the values decrease from 1 to 0, the curve falls rapidly, and for , it approaches the negative axis asymptotically. As the values increase from 1 to 10, the function increases monotonically from 0 to 1, and as values increase by a factor of 10 (for example, from 10 to 100) the function increases from 1 to 2. The same applies for the intervals , , and so on. Because the changes are very small for such large intervals, the curve can be well approximated by a straight line.

To switch bases, we let ; we will show that .

By definition, implies .

Taking the to the base of both sides gives .

Dividing by gives . Replacing by yields .