Chapter 6.4, Problem 7E

To determine

To Solve: The equation $2^x=64$ and check the solution.

Answer to Problem 7E

The solution of the equation $2^x=64$ is $x=6$

Explanation of Solution

Given information:

The equation $2^x=64$

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

  $\begin{array}{l}{2}^x=64\\ \text{Factor 64}\\ 2^x=2\times 2\times 2\times 2\times 2\times 2\\ \text{Using the property }{a}^n=a\times a\times a\times \mathrm{..................}n\text{ times}\\ 2^x=2^6\\ \text{Using the property If }{\left(a\right)}^n={\left(a\right)}^m\text{ then }m=n\\ x=6\end{array}$

To check put the value of $x=6$ in the expression

  $\begin{array}{l}{2}^x=64\\ 2^6\stackrel{?}{=}64\\ 64=64\\ \text{As LHS}=RHS\text{ therefore the solution is correct}\end{array}$

Conclusion:

The solution of the equation $2^x=64$ is $x=6$