log transformation for dummies

Question

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!

Answer 1

We can make use of the following property of $\log$: $$\log(a^b) = b\log(a)$$

So, in this case: $$y = -\log(x)$$ $$y = -\log(x^{1.5}) = -1.5\log(x)$$

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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: $$1.5(y) = -\log(x^{1.5})$$