# Do ALL curves get "straightened" by taking the log?

Try it yourself and find out...

So, although log transformation is powerful, it is not ALL-powerful. The only curves that actually get straightened are the ones generated by functions that include "some number to the power of the x variable". Why is that? Recall:

any population that GROWS OR SHRINKS BY A CONSTANT FACTOR will appear as a straight line when log-transformed.

And that's what doubling does -- it multiplies the population by a constant factor (2, that is) over a period of time. **But other mathematical functions, such as the ones above, involve some sort of change in which the factor of change is NOT constant.** Let's take the parabola, for example: y = x^{2}

x | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

y | 1 | 4 | 9 | 16 | 25 | 36 |

factor | 4 |
2.25 |
1.78 |
1.56 |
1.44 |

The factor by which y is increasing starts at 4, then drops to 2.25, then 1.78, and so on. Although we might think that a parabola grows "pretty fast", it is in fact "slow" compared to exponential growth that starts with the same factor.

As a general guideline:

**if growth factors are increasing**,

then growth is proceeding**faster than exponential**,

and that part of the**log-transformed graph will curve up**.**if growth factors are decreasing**,

then growth is proceeding**slower than exponential**,

and that part of the**log-transformed graph will have a flattening curve**.- Only
**constant growth factors**(exponential growth)

leads to a**straight-line**when the graph is log-transformed.

Copyright University of Maryland, 2007

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