Swing the Logarithmic Curve around (1, 0)
The logarithmic function to the base , where and , is defined by if and only if ; the domain is and the range is .[more]
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 .[less]
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 .