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How do I find the natural log of a fraction?

1 Answer

Dec 13, 2015

Apply the identity

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

Explanation:

Logarithms have the following useful properties:

$ln (a b) = ln (a) + ln (b)$ $ln(ab) = ln(a) + ln(b)$
$ln (a^{x}) = x ln (a)$ $ln(a^x) = xln(a)$

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

Applying these to a fraction, we get

$ln (\frac{a}{b}) = ln (a b^{- 1}) = ln (a) + ln (b^{- 1}) = ln (a) - ln (b)$ $ln(a/b) = ln(ab^(-1)) = ln(a) + ln(b^-1) = ln(a) - ln(b)$

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

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