Question

solve the equation ((128)^{x} = left( \frac{1}{2}
ight)^{2x - 3})
a. (x = \frac{1}{3})
b. (x = -\frac{3}{5})
c. (x = 5)
d. the equation has no solution

Explanation:

Step1: Express bases as powers of 2

We know that \(128 = 2^7\) and \(\frac{1}{2}=2^{-1}\). Substitute these into the equation:
\((2^7)^x=(2^{-1})^{2x - 3}\)

Step2: Apply exponent rules

Using the rule \((a^m)^n=a^{mn}\), we get:
\(2^{7x}=2^{-1\times(2x - 3)}\)
Simplify the exponent on the right:
\(2^{7x}=2^{-2x + 3}\)

Step3: Set exponents equal

Since the bases are the same and the equation holds, the exponents must be equal:
\(7x=-2x + 3\)

Step4: Solve for x

Add \(2x\) to both sides:
\(7x + 2x=-2x+3 + 2x\)
\(9x = 3\)
Divide both sides by 9:
\(x=\frac{3}{9}=\frac{1}{3}\)

Answer:

A. \(x = \frac{1}{3}\)