# 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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