Solving Equations: How to Handle 2^x - 7^{x-1} -2^{x 1} 7^x

Solving Equations: How to Handle 2x - 7x-1 -2x 1 7x

This article provides a detailed step-by-step guide to solving the exponential equation 2x - 7x-1 -2x 1 7x. It covers both real and complex solutions and explains the use of logarithms in finding the value of x. Let's dive into the solution.

Understanding the Equation

The equation in question is:

2x - 7x-1 -2x 1 7x

Step-by-Step Solution

Transforming the Equation

First, let us rewrite the equation in a form that groups terms with the same base together:

2x - 7x-1 -2x 1 7x We can rewrite the right side to make the exponents clearer:

2x - 7x-1 -2×2x 7×7x-1 This results in:

2x - 7x-1 -2x 1 7x

Isolating the Terms

Let's isolate the terms involving 2 and 7:

2x 2x 1 7x 7x-1

Combine the terms on the left side:

2x(1 2) 7x-1(7 1)

This simplifies to:

3×2x 8×7x-1

Using Logarithms

To solve for x, take the natural logarithm of both sides:

ln(3×2x) ln(8×7x-1)

Apply the logarithm rules:

ln(3) xln(2) ln(8) (x-1)ln(7)

Expanding the right side:

xln(2) - xln(7) ln(8) - ln(3) - ln(7)

Factor out x on the left side:

x(ln(2) - ln(7)) ln(8) - ln(3) - ln(7)

Solve for x:

x (ln(8) - ln(3) - ln(7)) / (ln(2) - ln(7))

Using the identities:

logab logcb / logca and logab - logac logab/c

we can express:

x log7/2(8/21)

Complex Solutions

The equation has real as well as complex solutions. For complex solutions, we have:

x log7/2(21/8) - 2iπn, where n ∈ Z

Conclusion

In summary, the equation 2x - 7x-1 -2x 1 7x is solved using logarithms. The step-by-step process involves isolating terms, transforming the equation, and using logarithmic identities to find the value of x. For complex solutions, we introduce the concept of imaginary numbers.

Related Keywords

exponential equations logarithms complex solutions