%5Ex%3D27.jpeg "(1/3)^x=27")
Solving the Exponential Equation: (1/3)^x = 27
This article explores how to solve the exponential equation (1/3)^x = 27. We will demonstrate the steps involved and provide a clear explanation of the underlying concepts.
Understanding the Equation
The equation (1/3)^x = 27 involves an unknown exponent, x. Our goal is to find the value of x that makes the equation true. To do this, we will utilize the properties of exponents and logarithms.
Solving the Equation
-
Express both sides with the same base: Since 27 is 3 cubed (3^3), we can rewrite the equation as: (1/3)^x = 3^3
-
Rewrite the left side with a positive exponent: Using the property (a/b)^n = a^n / b^n, we can rewrite the left side: (3^-1)^x = 3^3
-
Apply the power of a power rule: This rule states that (a^m)^n = a^(m*n). Applying this to our equation: 3^(-x) = 3^3
-
Equate the exponents: Since the bases are now the same, we can equate the exponents: -x = 3
-
Solve for x: Multiply both sides by -1 to isolate x: x = -3
Solution
Therefore, the solution to the equation (1/3)^x = 27 is x = -3.
Verification
To verify our solution, we can substitute x = -3 back into the original equation:
(1/3)^(-3) = 27
This simplifies to:
(3)^3 = 27
Which is true, confirming that our solution is correct.
Conclusion
Solving exponential equations like (1/3)^x = 27 requires understanding the properties of exponents and logarithms. By expressing both sides of the equation with the same base, we can then equate the exponents and solve for the unknown variable. This process provides a clear and methodical approach to finding the solution.