# How do I find the natural log of a fraction?

Dec 13, 2015

Apply the identity

$\ln \left(\frac{a}{b}\right) = \ln \left(a\right) - \ln \left(b\right)$

#### Explanation:

Logarithms have the following useful properties:

• $\ln \left(a b\right) = \ln \left(a\right) + \ln \left(b\right)$

• $\ln \left({a}^{x}\right) = x \ln \left(a\right)$

(As an exercise, try confirming these using the definition of a logarithm: $\ln \left(a\right) = x \iff a = {e}^{x}$)

Applying these to a fraction, we get

$\ln \left(\frac{a}{b}\right) = \ln \left(a {b}^{- 1}\right) = \ln \left(a\right) + \ln \left({b}^{-} 1\right) = \ln \left(a\right) - \ln \left(b\right)$

Thus if you can evaluate the logarithm of the numerator and of the denominator, you can evaluate the logarithm of the fraction.