Let's analyze and solve the equation [tex]\((\sqrt{5})^{x-1} = 25\)[/tex].
Step 1: Rewrite 25 as a power of 5
25 can be expressed as [tex]\(5^2\)[/tex].
So the equation becomes:
[tex]\[
(\sqrt{5})^{x-1} = 5^2
\][/tex]
Step 2: Express [tex]\(\sqrt{5}\)[/tex] as a power of 5
[tex]\(\sqrt{5}\)[/tex] can be written as [tex]\(5^{1/2}\)[/tex].
Substituting this into the equation, we get:
[tex]\[
(5^{1/2})^{x-1} = 5^2
\][/tex]
Step 3: Apply the power of a power property
The property [tex]\((a^m)^n = a^{mn}\)[/tex] allows us to combine the exponents. Applying this property gives:
[tex]\[
5^{(1/2) \cdot (x-1)} = 5^2
\][/tex]
Step 4: Equate the exponents
Since the bases are the same (both are base 5), we can set the exponents equal to each other:
[tex]\[
\frac{x-1}{2} = 2
\][/tex]
Step 5: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], we first multiply both sides of the equation by 2:
[tex]\[
x - 1 = 4
\][/tex]
Next, we add 1 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4 + 1
\][/tex]
[tex]\[
x = 5
\][/tex]
Conclusion:
The value of [tex]\(x\)[/tex] that satisfies the equation [tex]\((\sqrt{5})^{x-1} = 25\)[/tex] is:
[tex]\[
x = 5
\][/tex]
