# log transformation for dummies

I have a question which is probaly very simple to answer for most people here:

We have a formula:

y = -log(x)

Then this happens to x:

= -log(x^1.5)  or ( = -log(x^(15/10)) )

How do I now write up y?

Many thanks!

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use the formula : $\log(a^b)=b\log(a)$ (i hope the question was to rewrite the second expression in terms of $y$) –  Amire Jul 12 '13 at 13:15
@AmireBendjeddou Sorry for not being clear. Actually, I don't want to rewrite the second expression in terms of y, but I want to do the same to y as I did to x, while still y = -log(x). –  Rob Jul 12 '13 at 13:24
you made this trandsformation : $x \rightarrow x^{1.5 }$ so you are looking for $y^{1.5 }$?? (be more explicit please) –  Amire Jul 12 '13 at 13:27
Sorry was not thinking clearly, you are right. Now I can write it as 1.5y = -log(x^1.5). Thanks! –  Rob Jul 12 '13 at 13:41

We can make use of the following property of $\log$: $$\log(a^b) = b\log(a)$$
So, in this case: $$y = -\log(x)$$ $$y = -\log(x^{1.5}) = -1.5\log(x)$$
Note: I treated the above as if $y$ were a function of $x$, and we applied the transformation $x \mapsto x^{1.5}$. If this isn't a transform problem:
If $y = -\log(x)$, then: $$-\log(x^{1.5}) = -1.5\log(x) = 1.5(-\log(x)) = 1.5(y)$$ Thus: $$1.5(y) = -\log(x^{1.5})$$