A) \[\log \left( \frac{b}{a} \right)\]
B) \[\log \left( \frac{a}{b} \right)\]
C) \[\frac{a}{b}\]
D) \[\log {{a}^{b}}\]
Correct Answer: B
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{a}^{x}}-{{b}^{x}}}{x}=\underset{x\to 0}{\mathop{\lim }}\,\,\left( \frac{{{a}^{x}}-1}{x} \right)-\underset{x\to 0}{\mathop{\lim }}\,\,\left( \frac{{{b}^{x}}-1}{x} \right)\] \[=\log \,\,a-\log \,\,b=\log \,(a/b)\].
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