Solving for x in 2^x 2^x 2^x 192: A Step-by-Step Guide

In mathematics, exponential equations like 2^x 2^x 2^x 192 often require a systematic approach to determine the value of the unknown variable. This guide walks through one such example, breaking down each step in detail to help you understand the process thoroughly.

Understanding the Equation

The given equation is a cubic form of a power, which can be represented as 32^x 192. This is because (2^x) (2^x) (2^x) 2^{3x}, and therefore 2^{3x} 192.

Step-by-Step Solution

First, recognize that the product 2^x cdot 2^x cdot 2^x can be simplified as 32^x because 2^{3x} (2^x)^3 32^x. So, the equation simplifies to:

32^x 192

Next, take the natural logarithm (ln) of both sides to solve for x algebraically:

ln(32^x) ln(192)

x cdot ln(32) ln(192)

x frac{ln(192)}{ln(32)}

Alternatively, use your knowledge of exponents to directly solve the equation. Recognize that 64 2^6. Therefore, the equation becomes:

2^x 64

Since 64 2^6, we have:

2^x 2^6

This implies:

x 6

Conclusion

Thus, the value of x in the given equation is 6. This method can be applied to similar exponential equations by simplifying the expression and either using logarithms or recognizing powers of the base.

Additional Insights

Understanding the properties of exponents and logarithms is crucial in solving such problems. The following points shed additional light on the solution process:

Exponential expressions can be simplified by factoring or combining like terms.

The natural logarithm is a powerful tool for solving exponential equations.

Recognizing powers of the same base allows for straightforward solutions, such as 64 2^6.

By mastering these techniques, you can solve a variety of exponential equations efficiently and accurately.