Question

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

Answer 1

Answer:

$\log\left(\dfrac{\sqrt{x}\:z^2}{4y^2}\right)$

Step-by-step explanation:

$\dfrac12\log(x)-2\log(2y)+2\log(z)$

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

$\implies \log(x)^{\frac12}-\log(2y)^2+\log(z)^2$

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

$\implies \log(\sqrt{x})-\log(4y^2)+\log(z^2)$

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

$\implies \log\left(\dfrac{\sqrt{x}}{4y^2}\right)+\log(z^2)$

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

$\implies \log\left(\dfrac{\sqrt{x}\:z^2}{4y^2}\right)$