Solve the equation:

[tex]\[ 128 = 2^{3x - 5} \][/tex]

Which answer shows the correct steps for solving the equation?

A.
[tex]\[
\begin{aligned}
128 & = 2^{3x - 5} \\
2^7 & = 2^{3x - 5} \\
7 & = 3x - 5 \\
12 & = 3x \\
4 & = x
\end{aligned}
\][/tex]

B.
[tex]\[
128 = 2^{3x - 5} \\
4^4 = 2^{3x - 5} \\
4 = 3x - 5 \\
9 = 3x \\
3 = x
\][/tex]

C.
[tex]\[
128 = 2^{3x - 5} \\
2^7 = 2^{3x - 5} \\
7 = 3x - 5 \\
2 = 3x \\
\frac{2}{3} = x
\][/tex]

D.
[tex]\[
128 = 2^{3x - 5} \\
2 \cdot 4^3 = 2^{3x - 5} \\
2 \cdot 3 = 3x - 5
\][/tex]



Answer :

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]