Solving Equations of Exponential Form: Finding the Value of x in 4^(x1.5) * 9^x 6^(x1)

Understanding and solving exponential equations is a crucial skill in both academic and real-world contexts. This article will guide you through the process of solving the equation 4^(x1.5) * 9^x 6^(x1).

Introduction to Exponential Equations

Exponential equations often appear in scenarios where a quantity changes at a rate proportional to its current value. For instance, they are used in population growth, radioactive decay, and compound interest.

Step-by-Step Solution

1. Simplifying the Equation

We start with the given equation:

4^(x1.5) * 9^x 6^(x1)

This can also be written as:

4^x * 4^1.5 * 9^x 6^x * 6^1

Simplifying further:

4^x * 4^(3/2) * 9^x 6^x * 6^1

This simplifies to:

4^x * 9^x * 4^(3/2) 6^x * 6^1

Further simplification gives us:

(4 * 9)^x * 4^(3/2) 6^x * 6^1

This simplifies to:

36^x * 4^(3/2) 6^x * 6

Now, let's use the substitution method for easier solving:

2. Substituting Variables

Let y (3/2)^x.

The equation becomes:

36^x * 4^(3/2) 6^x * 6

Dividing by 4^x:

36 * (3/2)^2x 6 * 6^x / 4^x

This simplifies to:

36 * (3/2)^2x 6 * (3/2)^x

Rewriting the equation:

(36 / 6) * (3/2)^2x (3/2)^x

Further simplifies to:

6 * (3/2)^2x (3/2)^x

Let's denote t (3/2)^x.

The equation becomes:

6t^2 - 6t - 1 0

3. Solving the Quadratic Equation

Using the quadratic formula:

t (-b ± √(b^2 - 4ac)) / 2a

Here, a 6, b -6, and c -1.

Thus:

t (6 ± √((-6)^2 - 4 * 6 * (-1))) / (2 * 6)

t (6 ± √(36 24)) / 12

t (6 ± √60) / 12

t (6 ± 2√15) / 12

t (3 ± √15) / 6

This gives us:

t (3 √15) / 6 and t (3 - √15) / 6

Since t (3/2)^x, we can solve for x.

4. Solving for x

Using the definition of logarithms:

x ln(t) / ln((3/2))

Substituting the values of t:

x ln((3 √15) / 6) / ln((3/2)) and x ln((3 - √15) / 6) / ln((3/2))

Calculating the numerical values:

x ≈ 1.709511291 and x ≈ 3.419022583

Conclusion

Therefore, the solutions to the equation 4^(x1.5) * 9^x 6^(x1) are:

x ≈ 1.709511291 or x ≈ 3.419022583

Additional Tips

Always simplify the equation as much as possible before proceeding. Use substitutions to convert complex equations into simpler forms. Apply the quadratic formula to solve for variables. Remember to use logarithms to isolate the variable when dealing with exponential expressions.

Keywords

exponential equations, logarithms, algebraic solutions, math problem solving