Question

solve for x, log 15 root 5 to the base x = 2- log 3 root 5 to the base x

Answer 1

Dear Student,

$$\begin{gathered} {\log }_x\left(15\sqrt{5}\right)=2-{\log }_x\left(3\sqrt{5}\right) \\ \Rightarrow {\log }_x\left(15\sqrt{5}\right)+{\log }_x\left(3\sqrt{5}\right)=2 \\ \Rightarrow {\log }_x\left[\left(15\sqrt{5}\right)\left(3\sqrt{5}\right)\right]=2\left[\log \left(a\right)+\log \left(b\right)=\log \left(ab\right)\right] \\ \Rightarrow {\log }_x\left(225\right)=2 \\ \Rightarrow {\log }_x\left(225\right)={\log }_x\left(x^2\right)\left[a={\log }_b\left(b^a\right)\right] \\ \Rightarrow 225=x^2 \\ \Rightarrow x=\pm 15 \\ \Rightarrow x=15becausexcannotbenegative \end{gathered}$$
Regards,