Solving the Exponential Equation: 5^x + 3 = (1/125)^x
In this article, we will explore a fascinating exponential equation: 5^x + 3 = (1/125)^x. We will delve into the solution of this equation, discussing the steps to find the value of x.
Step 1: Simplify the Right-Hand Side
Let's start by simplifying the right-hand side of the equation. We know that 1/125 can be written as 5^(-3). Therefore, we have:
(1/125)^x = (5^(-3))^x
Using the property of exponents, we can rewrite this as:
5^(-3x)
So, the equation becomes:
5^x + 3 = 5^(-3x)
Step 2: Equate the Exponents
Since both sides of the equation have a base of 5, we can equate the exponents:
x = -3x
Step 3: Solve for x
Now, we can solve for x by dividing both sides by -3:
x = -x/3
Multiplying both sides by -3 to get rid of the fraction, we get:
-3x = x
Subtracting x from both sides gives us:
-4x = 0
Dividing both sides by -4, we finally get:
x = 0
Conclusion
Thus, the value of x is 0. This is the only solution to the exponential equation 5^x + 3 = (1/125)^x. We have successfully solved the equation by simplifying the right-hand side, equating the exponents, and solving for x.
