Function f(x) = ax is increasing on R, if

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a > 1 f\left( x ight) = a^x f'\left( x ight) = a^x \log a ext { Given: f(x) is increasing on R .} \Rightarrow f'\left( x ight) > 0 \Rightarrow a^x \log a > 0 \Rightarrow a^x > 0 \left( ext { Logarithmic function is defined for positive values of a } ight) ext { We know,} a^x \log a > 0 ext { It can be possible when} a^x > 0 ext { and } \log a > 0 or a^x < 0 and \log a < 0 \left( ext { Not possible, logarithmic function is defined for positive values of a } ight) \Rightarrow \log a > 0 \Rightarrow a > 1 ext { So,f (x) is increasing when }a> 1 .

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