If x is a real number such that \frac{4^{x+2} - 4^x}{15} = 2^{x+3}, what is the value of x… — GRE General Test

If $x$ is a real number such that $\frac{4^{x+2} - 4^x}{15} = 2^{x+3}$ , what is the value of $x$ ?

Factoring out

4^x

from the numerator gives

4^x(4^2 - 1) = 15 \cdot 4^x

. Dividing by 15 simplifies the left-hand side to

4^x

. Expressing

4^x

2^{2x}

allows setting

2^{2x} = 2^{x+3}

. Equating the exponents

2x = x + 3

yields

x = 3

Factor the numerator of the left-hand side

4^{x+2} - 4^x = 4^x(4^2 - 1) = 4^x(16 - 1) = 15 \cdot 4^x

Factoring out the common exponential term

4^x

simplifies the subtraction.

Simplify the fraction on the left-hand side

\frac{15 \cdot 4^x}{15} = 4^x

The factor of 15 in the numerator cancels with 15 in the denominator.

Rewrite

4^x

in terms of base 2

4^x = (2^2)^x = 2^{2x}

Both sides must have a common base to equate their exponents.

Set the exponential expressions equal and solve for

x

2^{2x} = 2^{x+3} \implies 2x = x + 3 \implies x = 3

Since the bases are equal and positive (base 2), their exponents must be equal.

Factoring common exponential terms and equating exponents with identical bases

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