A) \[\log \frac{a}{b}\]
B) \[\log \frac{b}{a}\]
C) \[\log ab\]
D) \[\log (a+b)\]
E) 0
Correct Answer: A
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{b}^{x}}}{{{e}^{x}}-1}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}{{\log }_{e}}a-{{b}^{x}}{{\log }_{e}}b}{{{e}^{x}}}\] \[={{\log }_{e}}a-{{\log }_{e}}b\] \[={{\log }_{e}}\frac{a}{b}\]
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