Chapter 9.5, Problem 63STP

To determine

To solve the equation and also check the solution

Answer to Problem 63STP

The solution of the equation is $x=3$

Explanation of Solution

Given information:

The given equation is,

  $2{\log }_{5}x={\log }_{5}9$

Formula used:

  $a\log n=\log n^a$ And ${\log }_{a}m-{\log }_{a}n={\log }_a\left(\frac{m}{n}\right)$

Calculation:

The equation can be solved as:

At first,

  $\begin{array}{l}2{\log }_{5}x={\log }_{5}9\\ \Rightarrow {\log }_5\left(x^2\right)={\log }_{5}9\left[\text{Use }a\log n=\log n^a\right]\end{array}$

Now,

  $\begin{array}{l}{\log }_5\left(x^2\right)={\log }_{5}9\\ 0={\log }_{5}9-{\log }_5\left(x^2\right)\\ 0={\log }_5\left(\frac{9}{x^2}\right)\\ 5^0=\frac{9}{x^2}\left[\text{Convert to exponential form}\right]\\ 1=\frac{9}{x^2}\\ x^2=9\\ x=\pm \sqrt{9}\\ x=\pm 3\end{array}$

However, only +3 works as a solution because the log of a negative number is non-defined.

Now, check the solution

  $\begin{array}{l}\text{Left-hand-side}={\log }_5\left(x^2\right)\\ ={\log }_5{\left(3\right)}^2\left[\text{Put the value of }x\right]\\ ={\text{log}}_{5}9=\text{ Right-hand-side}\left(\text{proved}\right)\end{array}$

Therefore, the solution is $x=3$