Problem 30 - Algebra 2

Question

Write each equation in logarithmic form. \(3^{3}=27\)

Step-by-Step Solution

Answer

\(\log_3{27} = 3\)

Step 1: Identify the Components

The equation given is in exponential form as \(3^3 = 27\). Recognize the base of the exponent, which is 3, the exponent itself, which is 3, and the result on the right side of the equation, which is 27.

Step 2: Apply the Logarithmic Definition

Convert the exponential equation to its logarithmic form. Recall that if \(b^y = x\), then \(\log_b{x} = y\). Here, \(b = 3\), \(y = 3\), and \(x = 27\).

Step 3: Write the Equation in Logarithmic Form

Using the logarithmic definition, transform the equation \(3^3 = 27\) into logarithmic form as \(\log_3{27} = 3\).

Key Concepts

Exponential Form

Exponential form is a way to express numbers using a base and an exponent. In the equation \(3^3 = 27\), the number 3 is the base and the number 3 is the exponent. This means that 3 is being multiplied by itself a total of 3 times, resulting in 27. Understanding exponential form is crucial because it helps simplify expressions and solve equations involving repeated multiplication. By mastering exponential form, you can easily handle larger numbers and work with them efficiently.

Logarithmic Definition

Logarithms are the inverse operation of exponents. The logarithmic definition states that if you have an equation in the form \(b^y = x\), you can express this as a logarithm: \(\log_b{x} = y\). Here’s how it works:

• \(b\) is the base of the exponent (and of the logarithm, too).
• \(y\) is the exponent, showing how many times the base is multiplied by itself.
• \(x\) is the result of the exponential equation.

By understanding this definition, you can reverse a power operation into a logarithmic one, which can be incredibly helpful for solving complex equations and grasping the relationships between numbers.

Base of the Exponent

The base of an exponent is a fundamental part of exponentiation and logarithms. In the equation \(3^3 = 27\), the base is 3. It is the number that is being multiplied repeatedly. Recognizing the base is essential because it plays a key role in determining the growth rate or decay rate in applications of exponential functions. In logs, the base indicates which multiplication table you are referring to. If you are familiar with different bases such as 10 (common logarithms) or \(e\) (natural logs), you can deepen your understanding of both exponential and logarithmic relationships.

Converting Equations

Converting equations from exponential form to logarithmic form, and vice versa, is a vital skill in mathematics. This conversion helps in solving for unknown variables and understanding the relationship between different mathematical operations.Here’s the simple process:

• Start with an exponential equation, like \(3^3 = 27\).
• Identify the base (3), exponent (3), and result (27).
• Use the logarithmic definition to change the equation: \(\log_3{27} = 3\).

This conversion makes it easier to solve problems involving growth and decay, and helps in comprehending complex mathematical models in real-life situations, such as finance and science.