Sure! Let's solve the equation [tex]\( 4^{2x-1} = 2^{x+1} \)[/tex] step-by-step.
1. Express [tex]\(4^{2x-1}\)[/tex] in terms of base 2:
- Recall that [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex]. Thus, [tex]\( 4^{2x-1} \)[/tex] becomes [tex]\((2^2)^{2x-1}\)[/tex].
2. Simplify the exponent expression:
- Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we have [tex]\((2^2)^{2x-1} = 2^{2 \cdot (2x-1)}\)[/tex].
- Simplify the exponent: [tex]\(2 \cdot (2x-1) = 4x - 2\)[/tex]. So, [tex]\( 4^{2x-1} = 2^{4x-2} \)[/tex].
3. Rewrite the original equation with the simplified exponents:
- Now the equation [tex]\( 4^{2x-1} = 2^{x+1} \)[/tex] becomes [tex]\( 2^{4x-2} = 2^{x+1} \)[/tex].
4. Set the exponents equal to each other:
- Since the bases are the same (both are base 2), we can equate the exponents directly:
[tex]\[ 4x - 2 = x + 1 \][/tex]
5. Solve the resulting linear equation for [tex]\(x\)[/tex]:
- Start by isolating [tex]\(x\)[/tex] on one side of the equation:
[tex]\[ 4x - 2 = x + 1 \][/tex]
- Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 4x - x - 2 = 1 \][/tex]
[tex]\[ 3x - 2 = 1 \][/tex]
- Add 2 to both sides:
[tex]\[ 3x = 3 \][/tex]
- Divide both sides by 3:
[tex]\[ x = 1 \][/tex]
Therefore, the solution to the equation [tex]\( 4^{2x-1} = 2^{x+1} \)[/tex] is [tex]\( x = 1 \)[/tex].
