If 4^x - 3^{x - 0.5} = 3^{x + 0.5} - 2^{2x-1}, what is the value of x? — SAT

If $4^x - 3^{x - 0.5} = 3^{x + 0.5} - 2^{2x-1}$ , what is the value of $x$ ?

Cevap: 1.5

The correct answer is

1.5

(or

\frac{3}{2}

)

By rewriting

4^x

2^{2x}

and grouping the base 2 and base 3 terms, the equation becomes

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

. Factoring out the variable terms gives

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

. Simplifying the coefficients results in

\frac{3}{2} \cdot 2^{2x} = \frac{4}{\sqrt{3}} \cdot 3^x

. Dividing both sides to separate the variables and constants yields

\frac{2^{2x}}{3^x} = \frac{8}{3\sqrt{3}}

. Representing both sides with the common base

\frac{4}{3}

gives

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

. Equating the exponents leads to

x = 1.5

Rewrite the base 4 term as a power of 2

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

Expressing exponential terms with prime bases simplifies finding relationships between them.

Group terms with like bases on opposite sides of the equation

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

Separating different base systems allows us to factor out common exponential functions.

Factor out the common exponential factors from each side

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

This isolates the constant coefficients from the exponential functions of

x

Simplify the constant terms on both sides of the equation

2^{2x}\left(\frac{3}{2}\right) = 3^x\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right) = 3^x\left(\frac{4}{\sqrt{3}}\right)

Reducing the coefficients makes it easier to combine the constant terms.

Divide both sides to group the exponential terms together

\frac{2^{2x}}{3^x} = \frac{4}{\sqrt{3}} \cdot \frac{2}{3} = \frac{8}{3\sqrt{3}}

This sets up the equation in the form of

f(x) = C

where

f(x)

has a combined base.

Express both sides of the equation with a common base of

\frac{4}{3}

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

Since

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

, the left side simplifies to

\left(\frac{4}{3}\right)^x

. The right side can be rewritten as

\frac{8}{3\sqrt{3}} = \frac{2^3}{3^{1.5}} = \frac{(4^{0.5})^3}{3^{1.5}} = \frac{4^{1.5}}{3^{1.5}} = \left(\frac{4}{3}\right)^{1.5}

Equate the exponents

x = 1.5

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

Solving exponential equations using base conversion, exponent rules, and factoring.

Soru