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I have the following logarithm to compute,

$$\log{2}+\log{ \left( \frac{p \left( x \right)}{p \left( x \right)+q \left( x \right)} \right)}$$

Then, in the solution,

$$\log{ \left( \frac{p \left( x \right)}{\frac{p \left( x \right)+q \left( x \right)}{2}} \right)}$$

How 2 got inside the log?

imnotmurtaza
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1 Answers1

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Recall that $\log\left(a\right)+\log\left(b\right)=\log\left(ab\right)$: $$\log\left(2\right)+\log\left(\frac{p\left(x\right)}{p\left(x\right)+q\left(x\right)}\right)=\log\left(\frac{2p\left(x\right)}{p\left(x\right)+q\left(x\right)}\right)\text{,}$$ which is equivalent to the expression given in the solution: $$\log{\left(\frac{p\left(x\right)}{\frac{p\left(x\right)+q\left(x\right)}{2}}\right)}=\log{\left(\frac{p\left(x\right)}{1}\div\frac{p\left(x\right)+q\left(x\right)}{2}\right)}=\log{\left(\frac{p\left(x\right)}{1}\times\frac{2}{p\left(x\right)+q\left(x\right)}\right)}$$ $$=\log\left(\frac{2p\left(x\right)}{p\left(x\right)+q\left(x\right)}\right)=\log\left(2\right)+\log\left(\frac{p\left(x\right)}{p\left(x\right)+q\left(x\right)}\right)$$

imnotmurtaza
  • 105