Chapter 6.6, Problem 7E

To determine

To calculate: .The equation for the value of $x$ .

Answer to Problem 7E

The required value of $x$ is $7$ .

Explanation of Solution

Given Information:

The given equation is $5^{x-3}={25}^{x-5}$ .

Formula Used Power rule of logarithmic function:

  ${\log }_a{x}^p=p{\log }_{a}x$

Where $a$ is the base, $x$ is the variable and $p$ is the power of variable.

Calculation:

Consider the equation is $5^{x-3}={25}^{x-5}$ .

The given equation can be rewrite as

  $\begin{array}{c}{5}^{x-3}={25}^{x-5}\\ 5^{x-3}={\left\{{\left(5\right)}^2\right\}}^{x-5}\\ 5^{x-3}=5^2{}^{\left(x-5\right)}\\ 5^{x-3}=5^{2x-10}\end{array}$

Take common logarithm with base $10$ of the above equation $5^{x-3}=5^{2x-10}$ .

  ${\log }_{10}{5}^{x-3}={\log }_{10}{5}^{2x-10}$

Now, use power rule of logarithm ${\log }_a{x}^p=p{\log }_{a}x$ in the above equation.

  $\left(x-3\right){\log }_{10}5=\left(x-5\right){\log }_{10}5$

Divide each side by ${\log }_{10}5$ in the above equations.

  $\begin{array}{c}\frac{\left(x-3\right){\log }_{10}5}{{\log }_{10}5}=\frac{\left(2x-10\right){\log }_{10}5}{{\log }_{10}5}\\ x-3=2x-10\\ x-2x=-10+3\\ -x=-7\end{array}$

Simplify the above equation.

  $\begin{array}{c}-x=-7\\ x=7\end{array}$

Hence, the required value of $x$ is $7$ .