Solving for X in the Equation 7x 49: A Step-by-Step Guide

Understanding how to solve for x in an equation like 7x 49 is crucial for anyone studying mathematics. This guide will show you multiple methods to solve for x, including logarithms and exponent properties.

Method 1: Using Logarithms

One reliable method involves using logarithms to isolate x. The steps are as follows:

Take the logarithm of both sides: log7(7x) log7(49) Using the property of logarithms, this simplifies to: x log7(7) log7(49) As log7(7) 1, we have: x log7(49) Finding that log7(49) 2, since 72 49, we conclude: x 2

Method 2: Directly Using Exponent Properties

A more straightforward approach is to recognize the relationship between the bases and exponents:

Recognize that 49 is a power of 7, i.e., 49 72. So, 7x 72. Since the bases are the same, we can equate the exponents: x 2.

Method 3: Natural Logarithms

Another method involves using the natural logarithm (ln) for a more generalized approach:

Taking the natural logarithm of both sides: ln(7x) ln(49) Using the property of logarithms, this simplifies to: x ln(7) ln(49) Since ln(49) ln(72) 2 ln(7), we have: x ln(7) 2 ln(7) Dividing both sides by ln(7), we get: x 2 Thus, the solution is x 2.

Why x 2?

Understanding why x equals 2 is fundamental. The equation 7x 49 essentially means finding the exponent to which the base 7 must be raised to get the result 49. Since 72 49, it follows that the exponent x must be 2.

Conclusion

By using logarithms, recognizing the base properties, or employing natural logarithms, we can verify that x 2 is the correct solution. This method not only solves the equation but also reinforces the understanding of how exponents and logarithms work together.

Remember, the key to solving such problems is recognizing the properties of exponents and applying the appropriate logarithmic rules. By practicing these steps, you can tackle more complex equations with confidence.

For further reading and more exercises, consider reviewing basic logarithmic and exponential functions. This knowledge will be beneficial in higher-level mathematics and real-world applications.