**Solving the Equation: 6^2x - 1 + 2*25^x - 0.5 = 16*30^x - 1**

In this article, we will solve the equation 6^2x - 1 + 2*25^x - 0.5 = 16*30^x - 1.

**Step 1: Simplify the equation**

First, let's simplify the equation by combining like terms:

6^2x - 1 + 2*25^x - 0.5 = 16*30^x - 1

Simplifying the equation, we get:

36^x - 1 + 50^x - 0.5 = 480^x - 1

**Step 2: Rewrite the equation in a simpler form**

Now, let's rewrite the equation in a simpler form by combining the exponential terms:

(36^x + 50^x) - 1.5 = 480^x - 1

**Step 3: Solve for x**

To solve for x, we can use the property of exponential functions that states a^x = b^x implies x = log(a)/log(b).

Let's assume that 36^x + 50^x = k, where k is a constant.

Then, we can rewrite the equation as:

k - 1.5 = 480^x - 1

Simplifying further, we get:

k = 480^x + 0.5

Now, we can use the property of exponential functions to solve for x:

x = log(k)/log(480) + log(0.5)/log(480)

Simplifying further, we get:

**x = (log(k) + log(0.5))/log(480)**

Therefore, the solution to the equation 6^2x - 1 + 2*25^x - 0.5 = 16*30^x - 1 is x = (log(k) + log(0.5))/log(480), where k is a constant.

**Conclusion**

In this article, we have solved the equation 6^2x - 1 + 2*25^x - 0.5 = 16*30^x - 1 using algebraic manipulations and the properties of exponential functions. The solution to the equation is x = (log(k) + log(0.5))/log(480), where k is a constant.