Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 216=6^{3} $$

Step-by-Step Solution

Answer

The logarithmic form is \(\log_6(216) = 3\).

Step 1: Identify the base, exponent, and result

In the given equation, we have an exponential expression, which is written in the form of \(b^e = R\), where \(b\) is the base, \(e\) is the exponent, and \(R\) is the result. Here, the base \(b\) is 6, the exponent \(e\) is 3, and the result \(R\) is 216.

Step 2: Use the logarithmic form of an exponential equation

In logarithmic form, the equation \(b^e = R\) is written as \(\log_b(R) = e\). This means the logarithm of \(R\) with base \(b\) is equal to the exponent \(e\).

Step 3: Substitute the values into the logarithmic form

Replace \(b\), \(R\), and \(e\) with the corresponding values from the equation. Here, replace \(b\) with 6, \(R\) with 216, and \(e\) with 3. The equation becomes \(\log_6(216) = 3\).

Key Concepts

Exponential Equations

Let's begin by understanding exponential equations. An exponential equation is a mathematical expression in which a constant base is raised to a variable exponent. The general form is given by \( b^e = R \), where:

• \(b\) is referred to as the base.
• \(e\) is the exponent, which can be a variable or a constant.
• \(R\) is the result or value obtained.

For example, in the equation \(6^3 = 216\), 6 is the base, 3 is the exponent, and 216 is the result. Exponential equations are significant because they represent rapid growth or decay in various contexts, such as population growth, radioactive decay, and compound interest.
To solve or rearrange exponential equations, we often use logarithms, which help us transform them into a more manageable form.

Logarithms

Logarithms are the inverse operation of exponentiation, providing a way to solve for exponents when the base and the result are known. The concept of logarithms answers the question: "To what power must the base be raised to yield a given number?" The logarithm of a number \( R \) to a given base \( b \) is denoted as \( \log_b(R) \), and it calculates to the exponent \( e \) such that \( b^e = R \).

In practice, they help make multiplications more manageable by converting them into additions. For the exponential equation \( 6^3 = 216 \), converting it into logarithmic form results in \( \log_6(216) = 3 \). This means that 6 raised to the power of 3 results in 216. Logarithms are powerful tools in mathematics, used extensively in various fields like engineering, physics, and computer science.
They are particularly useful for solving equations where the exponent is an unknown, simplifying the handling of data that spans large numerical ranges.

Base and Exponent Identification

Identifying the base and exponent is crucial when working with exponential equations and converting them into logarithmic form. Here are the key components to recognize:

• Base (\(b\)): This is the number that is multiplied by itself as many times as dictated by the exponent.
• Exponent (\(e\)): This is the power to which the base is raised. It indicates how many times the base is used as a factor.

For instance, in \( 6^3 = 216 \), 6 is the base, meaning it is the number being repeatedly multiplied. The exponent, 3, determines how many times the base is multiplied by itself. Recognizing these parts allows us to understand and manipulate exponential equations effectively, especially when converting them to logarithmic form.
The conversion requires placing the base as the base of the logarithm, the result as the input, and the exponent as the output, forming the equation \( \log_b(R) = e \). This makes analyzing exponential relationships more straightforward and easier to work with.