Solving the Equation: 5^(x-1) = 10^(x) * 2^(-x) * 5^(x+1)
In this article, we will solve the equation 5^(x-1) = 10^(x) * 2^(-x) * 5^(x+1) and find the value of x.
Step 1: Simplify the Right-Hand Side of the Equation
First, let's simplify the right-hand side of the equation:
10^(x) = (2 * 5)^x = 2^x * 5^x
So, the equation becomes:
5^(x-1) = 2^x * 5^x * 2^(-x) * 5^(x+1)
Step 2: Combine the Exponents
Next, let's combine the exponents of the 2 and 5:
5^(x-1) = 2^(x-x) * 5^(x+x+1) 5^(x-1) = 2^0 * 5^(2x+1) 5^(x-1) = 1 * 5^(2x+1) 5^(x-1) = 5^(2x+1)
Step 3: Equate the Exponents
Now, we can equate the exponents:
x - 1 = 2x + 1
Step 4: Solve for x
Subtract 2x from both sides:
-x - 1 = 1
Add 1 to both sides:
-x = 2
Multiply both sides by -1:
x = -2
Conclusion
The value of x that satisfies the equation 5^(x-1) = 10^(x) * 2^(-x) * 5^(x+1) is x = -2.
