To solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the given ordered pair [tex]\((2^x, 3^y) = (8, 81)\)[/tex], let's break down the steps:
1. Solving for [tex]\(x\)[/tex]:
[tex]\[
2^x = 8
\][/tex]
We need to express 8 as a power of 2. We know that:
[tex]\[
8 = 2^3
\][/tex]
Thus, we can rewrite the equation as:
[tex]\[
2^x = 2^3
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
x = 3
\][/tex]
2. Solving for [tex]\(y\)[/tex]:
[tex]\[
3^y = 81
\][/tex]
We need to express 81 as a power of 3. We know that:
[tex]\[
81 = 3^4
\][/tex]
Thus, we can rewrite the equation as:
[tex]\[
3^y = 3^4
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
y = 4
\][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[
x = 3 \quad \text{and} \quad y = 4
\][/tex]
