Chapter 6.4, Problem 53E

To determine

To solve the equation:

  $8^{x-2}=\sqrt{8}$

Answer to Problem 53E

  $x=\frac{5}{2}$

Explanation of Solution

Given:

  $8^{x-2}=\sqrt{8}$

Concept Used:

Two powers with the same positive base(not equals to 1), are equal if and only if their exponents are equal, that is:

  $b^x=b^y\iff x=y;\text{ where }b>0,\text{ and }b\ne 1$

Calculation:

To solve the equation:

  $8^{x-2}=\sqrt{8}$

First, simplify the equation, as

  $\begin{array}{c}{8}^{x-2}=\sqrt{8}\\ \Rightarrow 8^{x-2}=8^{\frac{1}{2}}\end{array}$

Now, since base of above powers are same.

Thus

  $\begin{array}{c}{8}^{x-2}=8^{\frac{1}{2}}\\ \Rightarrow x-2=\frac{1}{2}\\ \Rightarrow x=2+\frac{1}{2}\\ \Rightarrow x=\frac{4+1}{2}\\ \Rightarrow x=\frac{5}{2}\end{array}$

Thus, the solution of given equation is:

  $x=\frac{5}{2}$