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Solving the Equation: (0.01)^x - 1 = 1000
In this article, we will explore the solution to the equation (0.01)^x - 1 = 1000. This equation involves exponential functions, and we will use logarithmic functions to solve it.
Understanding the Equation
The given equation is:
(0.01)^x - 1 = 1000
To solve this equation, we need to isolate the variable x. Let's start by adding 1 to both sides of the equation to get:
(0.01)^x = 1001
Using Logarithmic Functions
To solve for x, we can use the logarithmic function. Recall that the logarithmic function is the inverse of the exponential function. In this case, we can take the logarithm base 10 of both sides of the equation:
log((0.01)^x) = log(1001)
Using the property of logarithms that states log(a^b) = b * log(a), we can rewrite the equation as:
x * log(0.01) = log(1001)
Solving for x
Now, we can solve for x by dividing both sides of the equation by log(0.01):
x = log(1001) / log(0.01)
Using a calculator or logarithmic tables, we find that:
log(1001) ≈ 3.0004 log(0.01) ≈ -2
Substituting these values, we get:
x ≈ 3.0004 / -2 x ≈ -1501.52
So, the solution to the equation (0.01)^x - 1 = 1000 is x ≈ -1501.52.
Conclusion
In this article, we solved the equation (0.01)^x - 1 = 1000 using logarithmic functions. We isolated the variable x and used the properties of logarithms to find the solution. The final answer is x ≈ -1501.52.
