Problem 23 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{3} 81=4 $$

Step-by-Step Solution

Answer

3^4 = 81

Step 1: Understand the Definition

The logarithmic equation \(\log_{3} 81 = 4\) represents the power to which the base, 3, must be raised to obtain the number 81.

Step 2: Recall Exponential Form

A logarithm \(\log_b a = c\) can be rewritten in exponential form as \(b^c = a\). This means the base \(b\) raised to the power \(c\) equals \(a\).

Step 3: Apply to Given Problem

Using the rule from Step 2, rewrite \(\log_{3} 81 = 4\) in exponential form: \(3^4 = 81\).

Step 4: Verify the Equation

Check that \(3^4\) equals 81. Since multiplying 3 by itself four times (\(3 \times 3 \times 3 \times 3\)) equals 81, the conversion to exponential form is correct.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where an unknown variable appears in the exponent. This is a key concept when working with logarithms, as they often involve converting between different forms of representation. For instance, when you have a logarithmic equation, like in our example,

• you can transform it into an exponential equation.
• This helps to make complex calculations simpler, by rearranging the equation into a form that is often more intuitive to solve or verify.

Exponential equations take the form:\[ b^c = a\]Here, \(b\) is the base, \(c\) is the exponent, and \(a\) is the result or value you get. The process of rewriting an equation in this way highlights the mathematical relationship that exists between these elements. This understanding is crucial because it allows for greater flexibility in approaching various mathematical problems. Recognizing this form can simplify the process of teaching or solving these kinds of equations and is especially helpful in fields such as algebra and calculus. It allows you to see not just the equation itself, but the dynamic relationships between its parts.

Logarithms

Logarithms are another fundamental concept in mathematics that represent the opposite operation of exponentiation. They answer the question: "To what exponent must a certain base be raised, to produce a given number?" In simpler terms, if you know the result and the base, a logarithm tells you the power.

• The basic notation is \(\log_b a = c\), where \(b\) is the base, \(a\) is the argument, and \(c\) represents the exponent.
• This is a tool that is exceptionally helpful for dealing with exponential growth, decay, or for bringing high powered numbers into more manageable forms.

Logarithms convert multiplication into addition, division into subtraction, and exponentiation into multiplication by using properties such as

• \(\log_b(mn) = \log_b m + \log_b n\)
• \(\log_b(m/n) = \log_b m - \log_b n\)
• \(\log_b(m^n) = n\cdot \log_b m\)

Understanding logarithms helps in solving exponential equations more fluently and serves as a bridge between arithmetic operations and algebraic manipulations. By becoming comfortable with this concept, you equip yourself with the ability to move smoothly between the realms of addition, multiplication, and exponentiation.

Base and Exponent Relationship

The relationship between base and exponent in mathematics is foundational, especially in topics involving logarithms and exponential equations. A base is a number that is multiplied by itself as many times as dictated by the exponent. This relationship can be thought of as a repeated multiplication operation.

• If the base is 3 and the exponent is 4, it means you multiply 3 by itself 4 times (\(3 \times 3 \times 3 \times 3 = 81\)).
• In logarithmic terms, this relationship answers the question of how many times the base must be multiplied by itself to achieve a certain number.

The power of this relationship lies in its simplicity. It turns complex multiplication problems into simple ones through the structure it provides. Understanding how to switch between the base-exponent form and its logarithmic counterpart is a key skill.For example, with logarithms:

• The equation \(\log_{3} 81 = 4\) asks, "3 to what power gives 81?"
• Conversely, the exponential form \(3^4 = 81\) confirms this relationship by showing the actual computation.

This relationship is pivotal not just in theoretical mathematical contexts but also in real-world problems involving exponential growth and decay, like calculating compound interest or understanding population growth trends.