Chapter 4 - Inverse, Exponential, and Logarithmic Functions - 4.4 Evaluation Logarithms and the Change-of-Base Theorem - 4.4 Exercises - Page 456: 2

The function is increasing over its domain.

If we differentiate the function, we get $\frac{1}{x\ln a}$. as a>1 so $\ln a$ > 0 The domain is the set of positive real numbers so x > 0. So the derivative is positive throughout its domain. Hence the function is increasing throughout its domain.