Multiply two logs

Question

How can we multiply two logs or how can we simplify them for example: $$\ln(x)*\ln(1-x)$$

Answer 1

You can not combine these terms. Only in the case $$\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)$$ and $$\ln(ab)=\ln(a)+\ln(b)$$ if $$a,b>0$$

Answer 2

They're already in the simplest form. So the expression can't be simplified.

But you could combine them if you didn't mind it getting more complicated. I don't recommend it unless you've got a special use for the result:

$$\ln(a)\cdot\ln(b)=\ln(a^{\ln(b)})=\ln(b^{\ln(a)})$$

So it's not been simplified, just complicated in two different ways. But for the fun of it, let's equate the two ways:

$$\ln(a^{\ln(b)})=\ln(b^{\ln(a)})$$

then exponentiate both sides to get a nice identity:

$$a^{\ln(b)}=b^{\ln(a)}$$

which isn't what you were trying to do, but is a nice side effect of playing with it.

Answer 3

$\ln(x)+\ln(1-x) = \ln(1-x^2)$ but

$$\ln(x)*\ln(1-x)$$ cannot be multiplied or simplified further other way round anymore.