Question

Fully condense the expression. Assume that all variables represent positive numbers. (log x -2 log y+ 3 log z) (10 points)

Answer 1

Answer:  The required condensed expression is $\log\dfrac{xz^3}{y^2}.$

Step-by-step explanation:  We are given to fully condense the following logarithmic expression assuming that all variables represent positive numbers :

$E=\log x-2\log y+3\log z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We will be using the following properties of logarithms :

$(i)~\log a^b=b\log a,\\\\(ii)~\log a+\log b=\log(ab),\\\\(iii)~\log a-\log b=\log\dfrac{a}{b}.$

Therefore, from expression (i), we get

$E\\\\=\log x-2\log y+3\log z\\\\=\log x-\log y^2+\log z^3\\\\=\log(xz^3)-\log y^2\\\\=\log\dfrac{xz^3}{y^2}.$

Thus, the required condensed expression is $\log\dfrac{xz^3}{y^2}.$