Solving Equations with Exponential Terms: A Detailed Guide and Example Problem

Exponential equations play a crucial role in diverse fields including science, engineering, and economics. This article will guide you through solving a specific type of exponential equation, providing a detailed step-by-step approach and an illustrative example.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, which can be challenging but also incredibly rewarding to solve. They are often encountered in practical scenarios like population growth models, radioactive decay, and financial calculations involving compound interest. Understanding and solving them is essential for many professionals and students.

Example Problem: Solving (4^x - 3^{x-frac{1}{2}} 3^{frac{x}{2}} - 2^{2x-1})

Let's tackle the equation (4^x - 3^{x-frac{1}{2}} 3^{frac{x}{2}} - 2^{2x-1}) step by step.

Step 1: Rewrite Exponential Terms

First, we rewrite the terms to express them in a more manageable form. We know that (4^x 2^{2x}) and (2^{2x-1} frac{2^{2x}}{2}).

Substituting these into the original equation:

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

Step 2: Express Terms in Terms of (3^x)

Next, express (3^{x-frac{1}{2}}) and (3^{frac{x}{2}}) in terms of (3^x).

(3^{x-frac{1}{2}} frac{3^x}{sqrt{3}})

(3^{frac{x}{2}} 3^x sqrt{3})

Substituting these into the equation:

(2^{2x} - frac{3^x}{sqrt{3}} 3^x sqrt{3} - frac{2^{2x}}{2})

Step 3: Eliminate Fractions by Multiplying by 2

Multiply the entire equation by 2 to eliminate the fraction:

(2 cdot 2^{2x} - 2 cdot frac{3^x}{sqrt{3}} 2 cdot 3^x sqrt{3} - 2^{2x})

This simplifies to:

(2^{2x 1} - frac{2 cdot 3^x}{sqrt{3}} 2 cdot 3^x sqrt{3} - 2^{2x})

Step 4: Rearrange and Combine Like Terms

Rearrange and combine like terms:

(2^{2x 1} - 2^{2x} 2 cdot 3^x sqrt{3} - frac{2 cdot 3^x}{sqrt{3}})

(2^{2x} - frac{8 cdot 3^x}{3} frac{8 cdot 3^x}{sqrt{3}} - 2^{2x})

Isolate the terms with (x):

(3 cdot 2^{2x} 2 cdot 3^x left(sqrt{3} - frac{1}{sqrt{3}}right))

(3 cdot 2^{2x} 2 cdot 3^x cdot frac{2}{sqrt{3}})

Step 5: Isolate (x)

Divide both sides by 3:

(2^{2x} frac{2 cdot 3^x cdot 2}{3sqrt{3}})

(2^{2x} cdot sqrt{3} frac{8 cdot 3^x}{3sqrt{3}})

Combine the equation:

(frac{2^{2x}}{3^x} frac{8}{3sqrt{3}})

Let (left(frac{2^2}{3}right)^x frac{8}{3sqrt{3}})

Take the logarithm of both sides:

(x cdot logleft(frac{4}{3}right) logleft(frac{8}{3sqrt{3}}right))

(x frac{logleft(frac{8}{3sqrt{3}}right)}{logleft(frac{4}{3}right)})

This gives the solution for (x). You can compute the value using a calculator for a numerical approximation.

Example Verification

Let's verify the solution by substituting (x frac{3}{2}) back into the original equation:

(3^{frac{3}{2} - frac{1}{2}} 2^{2 cdot frac{3}{2} - 1} 3^{frac{1}{2}} 2^{2 cdot frac{3}{2} - 1})

(3^1 2^2 3^{frac{1}{2}} 2^2)

(3 4 sqrt{3} 4)

(7 7)

This confirms that our solution is correct.