Question

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

${\log }_5\left(x\right)+{\log }_3\left(x\right)=1$

Short answer

The value of x satisfying the equation is$x=1.921$

Step-by-step solution

Given information

we are given an equation${\log }_5\left(x\right)+{\log }_3\left(x\right)=1$

Simplify and solve for x

We get,

$$\begin{gathered} \frac{\ln x}{\ln 5}+\frac{\ln x}{\ln 3}=1 \\ \ln 3\times \ln x+\ln 5\times \ln x=\ln 5\times \ln 3 \\ (\ln 3+\ln 5)\times \ln x=\ln 5\times \ln 3 \\ \ln x=\frac{\ln 5\times \ln 3}{\ln 3+\ln 5} \\ x=1.921 \end{gathered}$$