online edu.

1. If $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$, prove that $\log (a+c)+\log (a+c-2 b)=2 \log |a-c|$ . \begin{aligned} & \frac{2}{b}=\frac{1}{a}+\frac{1}{c} \\ & \therefore 2 a c=b(a+c) \quad \ldots \ldots .(1) \\ & \text { Now } \\ & \log (a+c)+\log (a+c-2 b) \\ & =\log (a+c)(a+c-2 b) \\ & =\log \left\{(a+c)^2-2 b(a+c)\right\} \\ & =\log \left\{(a+c)^2-4 a c\right\} \quad[\text { from (1)] } \\ & =\log (a-c)^2 \\ & =2 \log |a-c| \quad(\text { proved }) \end{aligned} 2. Prove that $a^{\log _{a^2}x} \times b^{\log _{b^2} y} \times c^{\log _{c^2}z}=\sqrt{ x y z}$. \begin{aligned} & \text{Solution:}\\ & a^{\log _{a^2}x} \times b^{\log _{b^2} y}\times c^{\log _{c^2}z}=P \\ & \Rightarrow \log _{a^2} x \log a+\log _{b^2}y \log b+\log _{c^2} z \log c=\log p \\ & \Rightarrow \frac{\log x}{\log a^2} \log a+\frac{\log y}{\log y^2} \log b+\frac{\log z}{\log c^2}\log c=\log p \\ & \Rightarrow \quad \frac{\log x}{2}+\frac{\log y}{2}+\frac{\log