Question

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:

${\log }_2\left(x\right)+{\log }_6\left(x\right)=3$

Short answer

The value of x satisfying the equation is$x=29.7$

Step-by-step solution

Given information

We are given an equation${\log }_2\left(x\right)+{\log }_6\left(x\right)=3$

Simplify and solve for x

We get,

$$\begin{gathered} {\log }_2\left(x\right)-{\log }_6\left(x\right)=3 \\ \frac{\ln x}{\ln 2}-\frac{\ln x}{\ln 6}=3 \\ \ln 6\times \ln x-\ln 2\times \ln x=3\times \ln 2\times \ln 6 \\ (\ln 6-\ln 2)\times \ln x=3\ln 2\times \ln 6 \\ \ln x=\frac{3\times \ln 2\times \ln 6}{\ln 6-\ln 2} \\ x=29.7 \end{gathered}$$