Answer: The required condensed expression is [tex]\log\dfrac{xz^3}{y^2}.[/tex]
Step-by-step explanation: We are given to fully condense the following logarithmic expression assuming that all variables represent positive numbers :
[tex]E=\log x-2\log y+3\log z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We will be using the following properties of logarithms :
[tex](i)~\log a^b=b\log a,\\\\(ii)~\log a+\log b=\log(ab),\\\\(iii)~\log a-\log b=\log\dfrac{a}{b}.[/tex]
Therefore, from expression (i), we get
[tex]E\\\\=\log x-2\log y+3\log z\\\\=\log x-\log y^2+\log z^3\\\\=\log(xz^3)-\log y^2\\\\=\log\dfrac{xz^3}{y^2}.[/tex]
Thus, the required condensed expression is [tex]\log\dfrac{xz^3}{y^2}.[/tex]
