Rewrite The Logarithmic Equation As An Exponential Equation (u > 0)

In the realm of mathematics, logarithmic and exponential equations are fundamental concepts that are closely intertwined. Understanding the relationship between these two forms of equations is crucial for solving a wide range of mathematical problems. This article delves into the process of rewriting a logarithmic equation into its equivalent exponential form, providing a comprehensive explanation and illustrative examples. Specifically, we will focus on rewriting the logarithmic equation $\log_3 u = 4$ as an exponential equation, assuming that $u > 0$.

Understanding Logarithmic and Exponential Forms

To effectively rewrite a logarithmic equation as an exponential equation, it is essential to grasp the fundamental definitions and relationships between these two forms. A logarithmic equation expresses the exponent to which a base must be raised to produce a given value. Conversely, an exponential equation expresses the relationship between a base, an exponent, and the resulting value.

The general form of a logarithmic equation is:

$$\log_b x = y$$

where:

• b is the base of the logarithm (a positive number not equal to 1)
• x is the argument of the logarithm (a positive number)
• y is the exponent to which the base b must be raised to obtain x

The equivalent exponential form of the logarithmic equation is:

$$b^y = x$$

This equation states that the base b raised to the power of y equals x. Understanding this equivalence is the key to converting between logarithmic and exponential forms.

Key Concepts and Definitions

Before we proceed with the rewriting process, let's reinforce some key concepts and definitions:

• Base: The base of a logarithm is the number that is raised to a power. In the logarithmic equation $\log_b x = y$, b is the base. The base must be a positive number not equal to 1.
• Argument: The argument of a logarithm is the value for which the logarithm is being calculated. In the logarithmic equation $\log_b x = y$, x is the argument. The argument must be a positive number.
• Exponent: The exponent is the power to which the base is raised. In the logarithmic equation $\log_b x = y$, y is the exponent.
• Logarithm: The logarithm is the exponent to which the base must be raised to obtain the argument. In the logarithmic equation $\log_b x = y$, y is the logarithm.

The Relationship Between Logarithmic and Exponential Forms

The logarithmic and exponential forms are inverse operations of each other. This means that they undo each other. The logarithmic equation $\log_b x = y$ can be rewritten as the exponential equation $b^y = x$, and vice versa. This inverse relationship is crucial for solving logarithmic and exponential equations.

Rewriting the Logarithmic Equation $\log_3 u = 4$

Now, let's apply the concept of rewriting logarithmic equations to the specific equation given: $\log_3 u = 4$. Our goal is to transform this logarithmic equation into its equivalent exponential form.

To rewrite the logarithmic equation $\log_3 u = 4$ as an exponential equation, we need to identify the base, argument, and exponent in the logarithmic form and then use them to construct the exponential form. In this equation:

• The base is 3.
• The argument is u.
• The exponent is 4.

Using the general form of the exponential equation, $b^y = x$, we can substitute the values from our logarithmic equation:

• b (base) = 3
• y (exponent) = 4
• x (argument) = u

Substituting these values into the exponential form, we get:

$$3^4 = u$$

This is the exponential form of the logarithmic equation $\log_3 u = 4$. It states that 3 raised to the power of 4 equals u.

Step-by-Step Conversion

To further clarify the conversion process, let's break it down into a step-by-step approach:

1. Identify the base, argument, and exponent in the logarithmic equation. In $\log_3 u = 4$, the base is 3, the argument is u, and the exponent is 4.
2. Write the general form of the exponential equation: $b^y = x$.
3. Substitute the values from the logarithmic equation into the exponential form. Replace b with 3, y with 4, and x with u.
4. Simplify the exponential equation, if necessary. In this case, $3^4 = u$ is already in its simplest form.

Solving for u

While the primary goal was to rewrite the logarithmic equation as an exponential equation, we can also solve for u in the exponential form. To do this, we simply evaluate $3^4$:

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Therefore, u = 81. This demonstrates how rewriting a logarithmic equation into exponential form can help solve for unknown variables.

Examples of Rewriting Logarithmic Equations

To solidify your understanding, let's explore additional examples of rewriting logarithmic equations as exponential equations:

Example 1:

Rewrite the logarithmic equation $\log_2 8 = 3$ as an exponential equation.

• Base: 2
• Argument: 8
• Exponent: 3

Exponential form: $2^3 = 8$

Example 2:

Rewrite the logarithmic equation $\log_{10} 100 = 2$ as an exponential equation.

• Base: 10
• Argument: 100
• Exponent: 2

Exponential form: $10^2 = 100$

Example 3:

Rewrite the logarithmic equation $\log_5 25 = 2$ as an exponential equation.

• Base: 5
• Argument: 25
• Exponent: 2

Exponential form: $5^2 = 25$

These examples illustrate the consistent process of identifying the base, argument, and exponent in the logarithmic form and then using them to construct the equivalent exponential form.

The Significance of Rewriting Equations

The ability to rewrite logarithmic equations as exponential equations, and vice versa, is a fundamental skill in mathematics. This skill is crucial for:

• Solving logarithmic and exponential equations: Rewriting equations can simplify the process of finding unknown variables.
• Understanding the relationship between logarithmic and exponential functions: The inverse relationship between these functions becomes clearer when you can easily convert between their forms.
• Graphing logarithmic and exponential functions: Knowing the equivalent forms can aid in visualizing and graphing these functions.
• Applying logarithms and exponentials in real-world scenarios: Many real-world phenomena, such as compound interest, radioactive decay, and sound intensity, are modeled using logarithmic and exponential functions. The ability to manipulate these equations is essential for solving practical problems.

Conclusion

Rewriting a logarithmic equation as an exponential equation is a straightforward process that relies on understanding the fundamental relationship between these two forms. By identifying the base, argument, and exponent in the logarithmic equation and applying the general form of the exponential equation, you can easily convert between the two forms. The specific example of rewriting $\log_3 u = 4$ as $3^4 = u$ demonstrates this process clearly. Mastering this skill is essential for success in various areas of mathematics and its applications. Understanding the concept of logarithms and exponentials is crucial, and this article has provided a comprehensive guide to rewriting logarithmic equations into their exponential counterparts. Remember, the key is to identify the base, argument, and exponent, and then apply the exponential form $b^y = x$. By practicing these steps, you'll become proficient in converting between these forms, which is a fundamental skill in mathematics. This skill will not only help you solve equations but also deepen your understanding of the relationship between logarithmic and exponential functions. The ability to rewrite logarithmic equations is a powerful tool that will serve you well in your mathematical journey. Mastering this conversion will unlock new possibilities in problem-solving and enhance your overall mathematical comprehension.