Solving the Exponential Equation 2^x=4^y=8^z and 1/2x+1/4y+1/8z=4
In this article, we will explore how to solve a complex exponential equation involving the numbers 2, 4, and 8. The equation is given by:
$2^x=4^y=8^z$
$\frac{1}{2}x+\frac{1}{4}y+\frac{1}{8}z=4$
Step 1: Simplify the Exponential Equation
Let's start by simplifying the exponential equation. We can rewrite the equation as:
$2^x=(2^2)^y=2^{2y}$
$2^x=(2^3)^z=2^{3z}$
Now, we can equate the exponents:
$x=2y=3z$
Step 2: Express y and z in terms of x
From the previous step, we can express y and z in terms of x:
$y=\frac{x}{2}$
$z=\frac{x}{3}$
Step 3: Substitute into the Linear Equation
Now, substitute the values of y and z into the linear equation:
$\frac{1}{2}x+\frac{1}{4}(\frac{x}{2})+\frac{1}{8}(\frac{x}{3})=4$
Step 4: Simplify the Linear Equation
Simplify the linear equation:
$\frac{1}{2}x+\frac{1}{8}x+\frac{1}{24}x=4$
Combine like terms:
$\frac{12}{24}x+\frac{3}{24}x+\frac{1}{24}x=4$
$\frac{16}{24}x=4$
Step 5: Solve for x
Solve for x:
$\frac{2}{3}x=4$
$x=4\cdot\frac{3}{2}$
$x=6$
Step 6: Find y and z
Now that we have found x, we can find y and z:
$y=\frac{x}{2}=\frac{6}{2}=3$
$z=\frac{x}{3}=\frac{6}{3}=2$
Conclusion
Therefore, the solution to the equation 2^x=4^y=8^z and 1/2x+1/4y+1/8z=4 is x=6, y=3, and z=2.
