-84-left(12%5E-x-right)-2-left(16%5E-x-right)%3D0-).jpeg "( 64 Left(9^ X Right)-84 Left(12^ X Right) 2 Left(16^ X Right)=0 )")
Solving the Equation: 64(9^x) - 84(12^x) + 2(16^x) = 0
In this article, we will explore the solution to the equation: 64(9^x) - 84(12^x) + 2(16^x) = 0.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand what each term represents.
- 64(9^x) is a term where 9 is raised to the power of x, and then multiplied by 64.
- -84(12^x) is a term where 12 is raised to the power of x, and then multiplied by -84.
- 2(16^x) is a term where 16 is raised to the power of x, and then multiplied by 2.
Simplifying the Equation
To simplify the equation, we can start by noticing that 9, 12, and 16 can be expressed as powers of 3 and 4.
- 9 = 3^2
- 12 = 3 * 4
- 16 = 4^2
Using these expressions, we can rewrite the equation as:
64((3^2)^x) - 84(3^x * 4^x) + 2(4^2x) = 0
Simplifying further, we get:
64(3^(2x)) - 84(3^x * 4^x) + 2(4^(2x)) = 0
Finding the Solution
To find the solution to the equation, we can start by noticing that the exponents of the terms are all multiples of x. This suggests that we can use logarithms to solve the equation.
Taking the logarithm of both sides of the equation, we get:
log(64(3^(2x))) - log(84(3^x * 4^x)) + log(2(4^(2x))) = 0
Simplifying further, we get:
2x log(3) - x log(3) - x log(4) + log(2) = 0
Combining like terms, we get:
x log(3) - x log(4) + log(2) = 0
Solving for x, we get:
x = log(2) / (log(3) - log(4))
Conclusion
In conclusion, we have successfully solved the equation 64(9^x) - 84(12^x) + 2(16^x) = 0. The solution involves using logarithms to simplify the equation and combining like terms to find the value of x.
