Sure, let's solve the equation [tex]\(\left(\frac{3}{2}\right)^x = \frac{27}{8}\)[/tex] step-by-step.
1. Express both sides with a common base:
- Notice that [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex] and [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. So, the equation becomes:
[tex]\[
\left(\frac{3}{2}\right)^x = \frac{3^3}{2^3}
-= \left(\frac{3}{2}\right)^3
\][/tex]
2. Since the bases are the same, equate the exponents:
- Because we now have the same base on both sides of the equation, we can set their exponents equal to each other:
[tex]\[
x = 3
\][/tex]
3. Verification:
- To verify the solution, we substitute [tex]\(x = 3\)[/tex] back into the original equation and check if both sides are equal:
[tex]\[
\left(\frac{3}{2}\right)^3 = \frac{27}{8}
\][/tex]
- Calculate [tex]\(\left(\frac{3}{2}\right)^3\)[/tex]:
[tex]\[
\left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}
\][/tex]
- Both sides are indeed equal, verifying that our solution is correct.
So, the solution to the equation [tex]\(\left(\frac{3}{2}\right)^x = \frac{27}{8}\)[/tex] is:
[tex]\[
x = 3
\][/tex]
After solving for [tex]\(x\)[/tex], you will find that [tex]\(x = 3\)[/tex].
Therefore, the values attained are [tex]\((3, 3.375, 3)\)[/tex].
