Solving for (3^x) when (2^x 3) through Logarithmic Manipulation

When attempting to solve mathematical equations involving logarithms and exponents, understanding how to manipulate the expressions is crucial. In this article, we will explore how to find the value of (3^x) given that (2^x 3). This involves using logarithmic identities and properties of exponents.

Introduction to Solving for (3^x)

Given the equation (2^x 3), our goal is to find the value of (3^x). This can be achieved through a series of steps involving logarithms and exponent rules. We'll walk through the process step-by-step, highlighting important properties and identities along the way.

Multiplying Both Sides by Logarithms

Starting with the given equation:

(2^x 3)

Taking the logarithm of both sides can help us isolate (x):

(x log_2{2} log_2{3})

Since "(log_2{2} 1)", the equation simplifies to:

(x frac{log_2{3}}{1} log_2{3})

Expressing (3^x) Using Logarithms

Now that we have (x log_2{3}), we can substitute this into (3^x) to get:

(3^x 3^{log_2{3}})

We know from the properties of logarithms that (a^{log_b{c}} c^{log_b{a}}). Thus, we can rewrite (3^{log_2{3}}) as:

(3^{log_2{3}} (2^{log_2{3}})^{log_2{3}})

Simplifying the expression further, we get:

(3^{log_2{3}} 2^{log_2{3} cdot log_2{3}} 2^{(log_2{3})^2})

Using Different Logarithm Bases

To convert the expression to a more common form, we can use the change of base formula (log_b{a} frac{log{a}}{log{b}}). Applying this to our expression:

(2^{log_2{3}} 3^{frac{1}{log_2{2}}} 3^1 3)

Thus, we can write:

(3^{log_2{3}} 3^{frac{log{3}}{log{2}}})

This represents the exact value, but for practical purposes, we can approximate it:

(3^{log_2{3}} approx 5.704)

Multiple Methods and Their Analogies

Let's explore another method:

Using Natural Logarithms

Starting from (2^x 3) and taking the natural logarithm of both sides:

(x ln{2} ln{3})

Thus:

(x frac{ln{3}}{ln{2}})

Substituting this into (3^x):

(3^x 3^{frac{ln{3}}{ln{2}}})

This is equivalent to:

(3^x (2^{ln{3}})^{frac{1}{ln{2}}})

Which simplifies to:

(3^x (2^{ln{3}})^{frac{1}{ln{2}}} 2^{ln{3} cdot frac{1}{ln{2}}} 2^{log_2{3}} 3)

Conclusion

The value of (3^x) when (2^x 3) is (2^{log_2{3}}), which evaluates to approximately (5.704). This solution demonstrates the power of logarithmic and exponential properties in solving complex mathematical problems. Understanding these principles can greatly enhance your ability to manipulate and solve equations involving logarithms and exponents.

Keywords: logarithm, exponent, logarithmic equations, base conversion, logarithmic identities, exponent properties, change of base formula, natural logarithm, common logarithm.