Laws of Logarithms Issue

Question

I have trouble with this maths assignment. Can you tell me what was wrong?

Thanks for your help!

https://www.dropbox.com/s/57ubtuslqjriqvd/zrongg.png

Answer 1

The Quotient Property of Logarithms states that for $X,Y,b\in \mathbb{R}^+, b\neq{1},$ $$\log_b{\frac{X}{Y}}=\log_bX-\log_bY$$ Let $X=x^2$, and $Y=yz^3$. Thus $$\log_b{\frac{x^2}{yz^3}}=\log_bx^2-\log_byz^3$$ The Product Property of Logarithms states virtually the same formula, except $$\log_b{XY}=\log_bX+\log_bY$$ Let $X=y$, and $Y=z^3$. Thus $$\log_b{yz^3}=\log_by+\log_bz^3$$ Therefore our entire expansion is $$\log_b{\frac{x^2}{yz^3}}=\log_bx^2-(\log_by+\log_bz^3)$$ Finally, the Power Property of Logarithms states that, for $c\in \mathbb{R}$ $$\log_bX^c=c\log_bX$$ Thus $$\log_b{\frac{x^2}{yz^3}}=\log_bx^2-\log_by-\log_bz^3=2\log_bx-\log_by-3\log_bz$$

Answer 2

$$\log_a\left(\dfrac{x^2}{yz^3}\right)=2\log_ax-\log_a(yz^3)=2\log_ax-\log_ay-3\log_az$$