Chapter 6.5, Problem 9E

To determine

To Solve: The equation $7^{x-5}={49}^x$ and check the solution.

Answer to Problem 9E

The solution of the equation $7^{x-5}={49}^x$ is $x=-5$

Explanation of Solution

Given information:

The equation $7^{x-5}={49}^x$

Concept and Formula Used:

Rule of exponent and addition and subtraction.

  $a^n=a\times a\times a\times \mathrm{..................}n\text{ times,If }{\left(a\right)}^n={\left(a\right)}^m\text{ then }m=n\&{\left(a^m\right)}^n=a^{mn}$

Calculation:

The given equation is

  $\begin{array}{l}{7}^{x-5}={49}^x\\ \text{Factor 49}\\ 7^{x-5}={\left(7\times 7\right)}^x\\ \text{Using the property }{a}^n=a\times a\times a\times \mathrm{..................}n\text{ times}\\ 7^{x-5}={\left(7^2\right)}^x\\ \text{Using the property }{\left(a^m\right)}^n=a^{mn}\\ 7^{x-5}={\left(7\right)}^{2x}\\ \text{Using the property If }{\left(a\right)}^n={\left(a\right)}^m\text{ then }m=n\\ x-5=2x\\ \text{Subtract }x\text{ from both sides}\\ x-5-x=2x-x\\ -5=x\\ \text{or }x=-5\end{array}$

To check put the value of $x=-5$ in the expression

  $\begin{array}{l}{7}^{x-5}={49}^x\\ 7^{-5-5}\stackrel{?}{=}{\left(49\right)}^{-5}\\ 7^{-10}\stackrel{?}{=}{\left(7^2\right)}^{-5}\\ 7^{-10}=7^{-10}\\ \text{As LHS}=RHS\text{ therefore the solution is correct}\end{array}$

Conclusion:

The solution of the equation $7^{x-5}={49}^x$ is $x=-5$