Properties of logarithms

Algebra Level 5

Let \(a,b\) be positive real numbers. Which of the following is not always true?

\[ (A) : a^{\ln{(b)}} = b^{\ln{(a)}}\]

\[ (B) : \ln{(a)} = \lim_{h \to 0} \dfrac{a^h - 1}{e^h - 1}\]

\[ (C) : \log_b(a) = \dfrac{\ln(a)}{\ln(b)}\]

\[ (D) : \ln{(a^b)} = b\times \ln(a)\]

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