From Förberedande kurs i matematik 1

Because \displaystyle a^{2}\sqrt{a}=a^{2}a^{\frac{1}{2}}=a^{2+\frac{1}{2}}=a^{\frac{5}{2}}, the logarithm law, \displaystyle b\lg a=\lg a^{b}, gives that

\displaystyle \log _{a}a^{2}\sqrt{a}=\log _{a}a^{\frac{5}{2}}=\frac{5}{2}\centerdot \log _{a}a=\frac{5}{2}\centerdot 1=\frac{5}{2},

where we have used that \displaystyle \log _{a}a=1.

NOTE: In this exercise, we assume, implicitly, that \displaystyle \text{a}>0\text{ } and \displaystyle \text{a}\ne \text{1}.