Exploring the Value of p^q When log_r p 2 and log_r q 3

In this article, we will delve into a mathematical problem involving logarithmic and exponential expressions. Specifically, we will explore the value of p^q given the conditions log_r p 2 and log_r q 3. This exploration will involve understanding the fundamental concepts of logarithms and exponentiation, and how these can be applied to solve the problem efficiently.

Understanding Logarithms and Exponentiation

Before we proceed, it's important to have a clear understanding of what logarithms and exponentiation represent. Logarithms are a way to express the power to which a base must be raised to produce a given number. For instance, if we say log_r p 2, it means that r^2 p. Similarly, if log_r q 3, it means that r^3 q.

Solving for p and q

Given the conditions:

log_r p 2 implies p r^2 log_r q 3 implies q r^3

Now, we need to find the value of p^q. Substituting the values of p and q from the expressions above, we get:

p^q (r^2)^{r^3}

Applying the Power Rule of Exponents

To simplify the expression further, we will use the power rule of exponents, which states that (a^m)^n a^{mtimes n}. Applying this rule to the expression, we get:

(r^2)^{r^3} r^{2times r^3} r^{2r^3}

Thus, the value of p^q is r^{2r^3}.

Conclusion

In summary, we have explored the problem of finding the value of p^q given the conditions log_r p 2 and log_r q 3. Through the application of logarithmic and exponential properties, we were able to derive that p^q r^{2r^3}. Understanding and manipulating these expressions is a crucial skill in advanced mathematics and practical applications, such as in computer science and engineering.

Related Keywords

- Logarithms

- Exponentiation

- Polynomial expressions

References

1. MathIsFun - Logarithms

2. Khan Academy - Logarithmic and Exponential Equations