If 4^{2x + 1} = 8^{x - 1}, what is the value of x? — SAT

If $4^{2x + 1} = 8^{x - 1}$ , what is the value of $x$ ?

-5

The correct answer is

-5

. By converting the bases of both sides to

2

, the equation becomes

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

. Equating the exponents gives

2(2x+1) = 3(x-1)

, which expands to

4x+2 = 3x-3

. Solving for

x

yields

x = -5

Express both bases as powers of 2

4 = 2^2

and

8 = 2^3

To solve an exponential equation, we need to rewrite the terms with a common base.

Substitute the bases back into the equation and apply exponent power rules

(2^2)^{2x + 1} = (2^3)^{x - 1} \Rightarrow 2^{2(2x + 1)} = 2^{3(x - 1)}

Applying the power of a power property,

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

, allows us to simplify the exponents.

Set the exponents equal to each other

2(2x + 1) = 3(x - 1)

Since the bases are identical and positive, their exponents must be equal.

Distribute and solve the linear equation for

x

4x + 2 = 3x - 3 \Rightarrow x = -5

Distributing the constants gives

4x + 2 = 3x - 3

. Subtracting

3x

from both sides gives

x + 2 = -3

, and subtracting

2

from both sides isolates the variable to find

x = -5

Solving exponential equations by expressing both sides with a common base

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