Solving the Equation: 65^x - 11/25^x + 0.5 - 65^x + 1 = 0
In this article, we will explore the solution to the equation 65^x - 11/25^x + 0.5 - 65^x + 1 = 0. This equation involves exponential functions and algebraic manipulations.
Simplifying the Equation
Let's start by simplifying the equation. We can begin by combining like terms:
65^x - 65^x = 0 (since both terms cancel out)
The equation then becomes:
-11/25^x + 0.5 + 1 = 0
Rearranging the Equation
Next, we can rearrange the equation to isolate the term with the exponent:
-11/25^x = -1.5
Multiplying Both Sides by -25^x
To eliminate the fraction, we can multiply both sides of the equation by -25^x:
11 = 1.5 * 25^x
Dividing Both Sides by 1.5
Now, we can divide both sides of the equation by 1.5:
11/1.5 = 25^x
Taking the Logarithm of Both Sides
To solve for x, we can take the logarithm of both sides of the equation:
log(11/1.5) = log(25^x)
Applying the Properties of Logarithms
Using the properties of logarithms, we can rewrite the equation as:
log(11/1.5) = x * log(25)
Solving for x
Finally, we can solve for x by dividing both sides of the equation by log(25):
x = log(11/1.5) / log(25)
Therefore, the solution to the equation 65^x - 11/25^x + 0.5 - 65^x + 1 = 0 is x = log(11/1.5) / log(25).
