properties of logarithms ln12-ln2=ln6

Question

I checked wolframalpha and it says that ln12-ln2=ln6. How? i tried to do: ln12=ln(2*2*3) which may be 2ln(2*3) (which is probably wrong). I need help.

EDIT: Ok, thanks. Actually i could have just searched logarithms properties on google(didn't think about it). Sorry for taking your time.

Answer 1

You need to remember that $$\ln a - \ln b = \ln \left(\frac ab\right)$$

Applying that here gives you $$\ln (12) - \ln(2) = \ln\left(\frac {12}{2}\right) = \ln(6)$$

Note, alternatively, that we can use the property $$\ln(ab) = \ln a + \ln b$$ as well.$$\ln(12) = \ln(2\cdot 6) = \ln(2) + \ln 6$$ So $$\ln(12) - \ln 2 = \ln 2 + \ln 6 -\ln 2 = \ln 6$$

Answer 2

Hint: Use the property $$\ln x-\ln y=\ln\frac xy.$$