A Logarithmic Expression

Applying laws of logarithms,

$5\log2 + \dfrac{3}{2}\log25 + \dfrac{1}{2}\log49 - \log28$

= $\log 2^5 + \log 25^{\frac{3}{2}} + \log 49^{\frac{1}{2}} - \log 28$

= $\log 32 + \log 125 + \log 7 -\log28$

= $\log\dfrac{32\times125\times7}{28}$

= $\log 1000$

= $\log_{10}1000$

= $\boxed{3}$

$\large \begin{aligned} x & = 5 \log 2 + \frac 32 \log 25 + \frac 12 \log 49 - \log 28 \\ & = \log \left(\frac {2^5 \times 25^\frac 32 \times 49^\frac 12}{28} \right) \\ & = \log \left(\frac {2^5 \times 5^{\cancel 2 \times \frac 3{\cancel 2}} \times 7^{\cancel 2 \times \frac 1{\cancel 2}}}{2^2 \times 7} \right) \\ & = \log \left(\frac {2^{\cancel 53} \times 5^3 \times \cancel 7}{\cancel {2^2} \times \cancel 7} \right) \\ & = \log \left(10^3 \right) \\ & = \boxed{3} \end{aligned}$