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(27/8)^x+7=(4/9)^-3x

3 min read Jun 16, 2024
(27/8)^x+7=(4/9)^-3x

Solving Exponential Equations: A Step-by-Step Guide

This article will guide you through the process of solving the exponential equation:

(27/8)^x + 7 = (4/9)^-3x

We will use a combination of algebraic manipulation and the power of logarithms to isolate the variable 'x'.

1. Simplifying the Equation

  • Expressing Fractions as Powers:
    • Rewrite (27/8) as (3/2)³ and (4/9) as (2/3)². This will make the equation easier to work with.
    • Our equation now becomes: (3/2)³^x + 7 = (2/3)²^-3x
  • Applying Power Rules:
    • Simplify the exponents: (3/2)^(3x) + 7 = (2/3)^(-6x)

2. Isolate the Exponential Term

  • Subtract 7 from both sides:
    • (3/2)^(3x) = (2/3)^(-6x) - 7

3. Using Logarithms

  • Take the logarithm of both sides: Choose a base that will simplify the equation. The natural logarithm (ln) is a good choice.
    • ln[(3/2)^(3x)] = ln[(2/3)^(-6x) - 7]
  • Apply the Logarithm Rule: ln(a^b) = b*ln(a)
    • 3x * ln(3/2) = ln[(2/3)^(-6x) - 7]

4. Solving for x

  • Isolate x:
    • x = ln[(2/3)^(-6x) - 7] / [3 * ln(3/2)]

Unfortunately, due to the presence of the constant '7', we cannot directly solve for 'x' using algebraic methods. This equation requires numerical methods or approximations to find the solution.

5. Numerical Methods for Approximation

  • Graphing Calculator: Plot the left-hand side and the right-hand side of the original equation. The x-coordinate of the intersection point represents the solution.
  • Iterative Methods: Methods like Newton-Raphson iteration can be used to find an approximate solution.

Important Note:

  • The equation may have multiple solutions, depending on the behavior of the functions involved.
  • Numerical methods provide approximations, not exact solutions.

By following these steps and using numerical methods, you can effectively solve exponential equations like this one.

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