Problem 21 - College Algebra

Question

For the following exercises, rewrite each equation in logarithmic form. $$n^{4}=103$$

Step-by-Step Solution

Answer

\(\log_n(103) = 4\)

Step 1: Understanding Exponential Form

The given equation is in exponential form: \( n^4 = 103 \). An exponential equation has the general form of \( b^e = a \), where \( b \) is the base, \( e \) is the exponent, and \( a \) is the result.

Step 2: Identifying the Components

In the equation \( n^4 = 103 \), the base \( b = n \), the exponent \( e = 4 \), and the result \( a = 103 \). These components will help us rewrite the equation in logarithmic form.

Step 3: Rewriting in Logarithmic Form

The logarithmic form of an exponential equation \( b^e = a \) is \( \log_b(a) = e \). Thus, for \( n^4 = 103 \), we rewrite it as \( \log_n(103) = 4 \).

Key Concepts

Exponential Form

Exponential form is a mathematical way to express repeated multiplication using a base and an exponent. This method is particularly useful for representing very large or small numbers efficiently. The general structure of an exponential equation is represented by \( b^e = a \), where:

• \( b \) is the "base," which is the number that will be multiplied.
• \( e \) is the "exponent," which indicates how many times the base is multiplied by itself.
• \( a \) is the "result," or the value obtained after applying the exponent to the base.

For example, in the expression \( 2^3 = 8 \), the base is 2, the exponent is 3, and the result is 8. This expresses that 2 is multiplied by itself three times, resulting in 8. Understanding exponential form is crucial to converting equations into other mathematical forms like logarithms.

Base and Exponent

Every exponential equation consists of a base and an exponent. These are essential components that dictate the power relationship between numbers in such equations. The base is the number being repeatedly multiplied, and the exponent tells us the frequency of the multiplication step. Let's delve into the problem \( n^4 = 103 \).

• Here, \( n \) is the "base," indicating the number we will multiply four times.
• The number 4 is the "exponent," specifying the multiplication frequency.

To solve or manipulate equations, identifying bases and exponents is key. Once recognized, we can convert or simplify equations using logarithmic form or other mathematical expressions. This concept is fundamental in various fields such as physics, engineering, and finance, where understanding growth and change is crucial.

Equation Conversion

Converting equations from exponential form to logarithmic form involves understanding the relationship between the two representations. This conversion is an important process in mathematics as it provides insights into solving complex equations and understanding their properties. Given an exponential equation \( b^e = a \), we convert it to logarithmic form as \( \log_b(a) = e \). In essence, the conversion states "the exponent \( e \) is equal to the power to which the base \( b \) must be raised to obtain \( a \)."For example, from the exponential equation \( n^4 = 103 \), we perform the conversion:

• The base \( n \) becomes the base of the logarithm.
• The result 103 becomes the argument of the logarithm.
• The exponent 4 becomes the outcome of the logarithmic equation.

Thus, \( \log_n(103) = 4 \) is the equivalent logarithmic form. This conversion is valuable in problem-solving, especially in sciences where logarithms characterize exponential growth or decay.

Logarithms in Mathematics

Logarithms are a fundamental concept in mathematics, providing a bridge between exponential equations and arithmetic calculations. They allow us to simplify the multiplication of large numbers into the addition of smaller, more manageable numbers. Logarithms are key in various applications, ranging from scientific calculations to finance and computer science.In the context of our exercise, converting the exponential equation \( n^4 = 103 \) to its logarithmic form \( \log_n(103) = 4 \) demonstrates how logarithms can simplify and solve equations. Logarithmic equations express the power (or exponent) to which a base must be raised to produce a given number.Some benefits of using logarithms include:

• Simplification of complex calculations.
• Ease in solving exponential growth or decay problems.
• Application in computational algorithms for efficient data processing.

By understanding logarithms, students gain a versatile tool that aids in a wide array of mathematical and real-world problems, making them a critical component of modern mathematical education.