To solve the equation [tex]\( 128 = 2^{3x - 5} \)[/tex], we can follow these steps:
1. Express [tex]\( 128 \)[/tex] as a power of 2:
[tex]\( 128 \)[/tex] can be written as [tex]\( 2^7 \)[/tex], since [tex]\( 2^7 = 128 \)[/tex]. This gives us:
[tex]\[
128 = 2^{3x - 5}
\][/tex]
2. Rewrite the equation using the property of exponents that if the bases are equal, then the exponents must be equal:
[tex]\[
2^7 = 2^{3x - 5}
\][/tex]
3. Set the exponents equal to each other:
[tex]\[
7 = 3x - 5
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
First, isolate [tex]\( 3x \)[/tex] by adding 5 to both sides of the equation:
[tex]\[
7 + 5 = 3x
\][/tex]
Simplifying, we get:
[tex]\[
12 = 3x
\][/tex]
5. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{12}{3} = 4
\][/tex]
Therefore, the correct steps to solve the equation [tex]\( 128 = 2^{3x - 5} \)[/tex] are as follows:
1. [tex]\( 128 = 2^{3x - 5} \)[/tex]
2. [tex]\( 2^7 = 2^{3x - 5} \)[/tex]
3. [tex]\( 7 = 3x - 5 \)[/tex]
4. [tex]\( 12 = 3x \)[/tex]
5. [tex]\( 4 = x \)[/tex]
The correct answer among the given options is:
[tex]\[
\begin{aligned}
128 & = 2^{3x - 5} \\
2^7 & = 2^{3x - 5} \\
7 & = 3x - 5 \\
12 & = 3x \\
4 & = x
\end{aligned}
\][/tex]
