How to Solve the Equation 8<sup>x</sup> + 2<sup>x</sup> = 130?"

Solving the Equation: 8^x + 2^x = 130

We are tasked with solving the equation:

8^x + 2^x = 130

We begin by expressing 8^x as a power of 2. Since 8 = 2³, we can rewrite the equation as:

                (2³)^x + 2^x = 130

This simplifies to:

                2^3x + 2^x = 130

Let y = 2^x. Then, the equation becomes:

                y³ + y = 130

Now, we need to solve the cubic equation:

                y³ + y - 130 = 0

We can try some values for y to find a root. If we substitute y = 5, we get:

                5³ + 5 = 125 + 5 = 130

So, y = 5 is a solution. Since y = 2^x, we now have:

                2^x = 5

Taking the logarithm of both sides gives:

                x = log₂(5)

Using the change of base formula:

                x = log(5) / log(2)

Using a calculator:

                x ≈ 2.32

Thus, the solution to the equation is approximately:

x ≈ 2.32

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