Solving the Equation 2^x=3^y=6^-z
Given the equation 2^x=3^y=6^-z, we can solve for the value of 1/x + 1/y + 1/z.
Step 1: Simplify the Equation
First, let's simplify the equation by rewriting 6^-z as (2^2)^-z = 2^-2z. Now, we can equate the three expressions:
2^x = 3^y = 2^-2z
Step 2: Equate the Exponents
Since the bases are different, we can equate the exponents:
x = y(log3(2)) = -2z
Step 3: Find the Relationship between x, y, and z
From the above equation, we can see that x, y, and z are related by:
x + 2z = 0 ... (1) y = x / log3(2) ... (2)
Step 4: Find the Value of 1/x + 1/y + 1/z
Now, we can find the value of 1/x + 1/y + 1/z by substituting the values of x, y, and z:
1/x + 1/y + 1/z = 1/x + log3(2) / x + (-1/2) / x = (1 + log3(2) - 1/2) / x
Simplifying the expression, we get:
1/x + 1/y + 1/z = (log3(4)) / x
Conclusion
Therefore, the value of 1/x + 1/y + 1/z is (log3(4)) / x.
