Question

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

${\log }_{9}x+3{\log }_{3}x=14$

Short answer

The value of x satisfying the equation is$x=81$

Step-by-step solution

Given information

We are given an equation${\log }_{9}x+3{\log }_{3}x=14$

Change the base

We get,

$$\begin{gathered} {\log }_{9}x+3{\log }_{3}x=14 \\ \frac{{\log }_3\left(x\right)}{{\log }_{3}9}+3{\log }_{3}x=14 \\ \frac{{\log }_3\left(x\right)}{2}+3{\log }_{3}x=14 \\ {\log }_3\left(x\right)+6{\log }_{3}x=28 \\ 7{\log }_{3}x=28 \\ {\log }_{3}x=4 \end{gathered}$$

use the formula y=logax⇔x=ay

We get,

$$\begin{gathered} {\log }_3\left(x\right)=4 \\ x=3^4 \\ x=81 \end{gathered}$$