Let's solve the equation [tex]\((\sqrt{5})^{x-1}=25\)[/tex] step-by-step.
1. Rewrite the equation with base analysis:
We start with the given equation:
[tex]\[
(\sqrt{5})^{x-1}=25
\][/tex]
First, notice that [tex]\(\sqrt{5}\)[/tex] can be written as [tex]\(5^{1/2}\)[/tex]. So the equation becomes:
[tex]\[
(5^{1/2})^{x-1}=25
\][/tex]
2. Combine the exponents:
Using the property of exponents [tex]\((a^{m})^n = a^{m \cdot n}\)[/tex], we can rewrite the left-hand side:
[tex]\[
5^{(1/2)(x-1)} = 25
\][/tex]
Simplify the exponent on the left-hand side:
[tex]\[
5^{(x-1)/2} = 25
\][/tex]
3. Express the right-hand side with the same base:
Notice that [tex]\(25\)[/tex] is a power of 5:
[tex]\[
25 = 5^2
\][/tex]
So our equation now looks like:
[tex]\[
5^{(x-1)/2} = 5^2
\][/tex]
4. Set the exponents equal to each other:
Since the bases are the same and the equation is equal, we can set the exponents equal to each other:
[tex]\[
\frac{x-1}{2} = 2
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], first eliminate the fraction by multiplying both sides by 2:
[tex]\[
x-1 = 4
\][/tex]
Finally, add 1 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
So, the solution to the equation [tex]\((\sqrt{5})^{x-1}=25\)[/tex] is:
[tex]\[
x = 5
\][/tex]
