Find the real value of x that satisfies the exponential equation 4^x - 3^{x - \frac{1}{2}}… — JAMB UTME

Find the real value of $x$ that satisfies the exponential equation $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$ .

Answer: 1.5

The value of

x

1.5

(or

\frac{3}{2}

By using index laws to group base-2 terms on the left side and base-3 terms on the right side, we obtain

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

. Rearranging gives

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

. Equating indices gives

x = 1.5

Group like exponential terms with base 2 and base 3 on opposite sides of the equation.

4^x + 2^{2x - 1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}}

Grouping terms with common prime bases allows for factoring exponential terms.

Apply the product and power laws of indices:

4^x = 2^{2x}

2^{2x-1} = 2^{-1} \cdot 2^{2x}

3^{x \pm \frac{1}{2}} = 3^x \cdot 3^{\pm \frac{1}{2}}

2^{2x} + \frac{1}{2} \cdot 2^{2x} = 3^x \cdot \sqrt{3} + 3^x \cdot \frac{1}{\sqrt{3}}

Separating the variable exponents from constant exponents prepares each side for factoring.

Factor out

2^{2x}

from the left side and

3^x

from the right side, then simplify arithmetic terms.

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

Factoring isolates the variable terms

2^{2x}

and

3^x

from numerical constants.

Divide to form the ratio

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

and simplify the numerical fraction on the right.

\left(\frac{4}{3}\right)^x = \frac{8}{3\sqrt{3}}

Expressing both sides with unified variable bases facilitates solving for

x

by equating powers.

Rewrite

\frac{8}{3\sqrt{3}}

as a power of

\frac{4}{3}

and solve for

x

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{\frac{3}{2}} \implies x = \frac{3}{2} = 1.5

Since

\frac{8}{3\sqrt{3}} = \frac{4^{3/2}}{3^{3/2}} = (\frac{4}{3})^{3/2}

, equating exponents yields

x = 1.5

Solving mixed-base exponential equations by grouping, factoring, and converting to a unified base ratio.

Estimated Time:3m 0s

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