Solving the Equation: 3125^x - 1 = 625^2 x - 2
In this article, we will explore how to solve the equation 3125^x - 1 = 625^2 x - 2.
Understanding the Equation
The equation 3125^x - 1 = 625^2 x - 2 involves exponential functions with different bases and powers. To solve this equation, we need to use the properties of exponents and logarithms.
Simplifying the Equation
Let's start by simplifying the equation:
3125^x - 1 = 625^2 x - 2
Step 1: Write 3125 as a power of 5:
3125 = 5^5
So, the equation becomes:
(5^5)^x - 1 = 625^2 x - 2
Step 2: Write 625 as a power of 5:
625 = 5^4
Now, the equation becomes:
(5^5)^x - 1 = (5^4)^2 x - 2
Step 3: Use the property of exponents that states a^(mn) = (a^m)^n:
(5^5x) - 1 = (5^8)x - 2
Solving for x
Now, we can solve for x by equating the exponents:
5x = 8
x = 8/5
x = 1.6
Therefore, the value of x is 1.6.
Conclusion
In this article, we solved the equation 3125^x - 1 = 625^2 x - 2 by using the properties of exponents and logarithms. We simplified the equation by writing the bases as powers of 5 and then solved for x by equating the exponents. The final answer is x = 1.6.
