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Solving the Equation: (0.125)^x + 4 = 0
In this article, we will explore the solution to the equation (0.125)^x + 4 = 0. This equation involves exponentiation and requires some clever manipulation to arrive at the solution.
Understanding the Equation
The equation (0.125)^x + 4 = 0 is a simple-looking equation, but it requires some careful analysis to solve. The equation involves raising 0.125 to the power of x, which is a fraction, and then adding 4 to the result. Our goal is to find the value of x that satisfies the equation.
Rewriting the Equation
To start, let's rewrite the equation in a more familiar form. We can rewrite 0.125 as a fraction:
0.125 = 1/8
So, the equation becomes:
(1/8)^x + 4 = 0
Taking the Logarithm
To solve this equation, we can take the logarithm of both sides. This will allow us to isolate the exponent x. Let's take the natural logarithm (ln) of both sides:
ln((1/8)^x + 4) = ln(0)
Using the property of logarithms that allows us to bring down the exponent, we get:
x * ln(1/8) + ln(4) = 0
Simplifying the Equation
We can further simplify the equation by evaluating the logarithms:
x * (-ln(2)) + ln(4) = 0
Solving for x, we get:
x = ln(4) / ln(2)
x = 2
The Solution
Therefore, the solution to the equation (0.125)^x + 4 = 0 is x = 2.
In conclusion, solving the equation (0.125)^x + 4 = 0 requires careful manipulation of the equation and the use of logarithms. By following these steps, we can arrive at the solution x = 2.
