If 4^{x - 1} = 8, what is the value of x? — SAT

If $4^{x - 1} = 8$ , what is the value of $x$ ?

Answer: 2.5

2.5 (or 5/2)

The correct answer is 2.5 (or 5/2). By expressing 4 as

2^2

and 8 as

2^3

, the equation is rewritten as

(2^2)^{x-1} = 2^3

. Applying the exponent power rule simplifies this to

2^{2x-2} = 2^3

. Equating the exponents yields the linear equation

2x-2 = 3

. Solving for

x

gives

2x = 5

, which results in

x = 2.5

5/2

Rewrite both sides of the equation with a common base of 2.

(2^2)^{x-1} = 2^3

Expressing both bases as powers of 2 allows us to equate the exponents later.

Apply the power of a power exponent rule

(a^m)^n = a^{mn}

to the left side.

2^{2x-2} = 2^3

Multiplying the exponent 2 by the exponent

(x-1)

simplifies the expression to

2(x-1) = 2x-2

Equate the exponents and solve the linear equation for

x

x = 2.5

Since the bases are identical, their exponents must be equal, giving

2x-2 = 3

, which simplifies to

2x = 5

Solving exponential equations by finding a common base.

Question