Question

Express as single logarithm.
1/2 log (x) - 2 log (2y) + 3 log (z)

Answer 1

`\log\left((√(x)\:z^2)/(4y^2)\right)`

Explanation:

`\frac12\log(x)-2\log(2y)+2\log(z)`

`\textsf{Apply log power rule}: \quad n \log(x)=log(x)^n`

`\implies \log(x)^(\frac12)-\log(2y)^2+\log(z)^2`

`\textsf{Apply exponent rule}: \quad(ab)^c=a^cb^c`

`\implies \log(√(x))-\log(4y^2)+\log(z^2)`

`\textsf{Apply log quotient rule}: \quad \log(x)-\log(y)=log((x)/(y))`

`\implies \log\left((√(x))/(4y^2)\right)+\log(z^2)`

`\textsf{Apply log product rule}: \quad \log(x)+\log(y)=log(xy)`

`\implies \log\left((√(x)\:z^2)/(4y^2)\right)`