To solve the equation [tex]\(4^{x+1} = 8^x\)[/tex], we need to express both sides of the equation using the same base.
1. Expressing with Base 2:
Notice that [tex]\(4\)[/tex] and [tex]\(8\)[/tex] can both be written as powers of [tex]\(2\)[/tex]:
[tex]\[
4 = 2^2
\][/tex]
[tex]\[
8 = 2^3
\][/tex]
2. Rewrite the Equation:
Substitute [tex]\(4\)[/tex] and [tex]\(8\)[/tex] with [tex]\(2^2\)[/tex] and [tex]\(2^3\)[/tex], respectively:
[tex]\[
4^{x+1} = (2^2)^{x+1}
\][/tex]
[tex]\[
8^x = (2^3)^x
\][/tex]
3. Simplify the Exponents:
Use the properties of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(2^2)^{x+1} = 2^{2(x+1)}
\][/tex]
[tex]\[
(2^3)^x = 2^{3x}
\][/tex]
So the equation now looks like:
[tex]\[
2^{2(x+1)} = 2^{3x}
\][/tex]
4. Equate the Exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2(x+1) = 3x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Simplify the equation:
[tex]\[
2x + 2 = 3x
\][/tex]
Isolate [tex]\(x\)[/tex] by subtracting [tex]\(2x\)[/tex] from both sides:
[tex]\[
2 = x
\][/tex]
So, the solution to the equation [tex]\(4^{x+1} = 8^x\)[/tex] is:
[tex]\[
x = 2
\][/tex]
