Problem 2 - Intermediate Algebra

Question

For Problems \(1-10\), write each of the following in logarithmic form. For example, \(2^{3}=8\) becomes \(\log _{2} 8=3\) in logarithmic form. $$ 3^{3}=27 $$

Step-by-Step Solution

Answer

The logarithmic form is \(\log_{3} 27 = 3\).

Step 1: Identify Base, Exponent, and Result

The given equation is \(3^{3} = 27\). Here, the base is 3, the exponent is 3, and the result is 27.

Step 2: Understand Logarithmic Form

The logarithmic form of an equation \(b^{e} = n\) is written as \(\log_{b} n = e\), where \(b\) is the base, \(e\) is the exponent, and \(n\) is the result.

Step 3: Convert to Logarithmic Form

Following the format from Step 2, convert \(3^{3} = 27\) to logarithmic form: \(\log_{3} 27 = 3\).

Key Concepts

Exponents

Exponents are a fundamental part of mathematics. In an expression like \(3^3 = 27\), the number 3 is referred to as the "base," and the other 3 is the "exponent." Together, they form a "power." To put it simply, exponents tell us how many times to multiply the base number by itself. Here, \(3^3\) means \(3 \times 3 \times 3\), which equals 27. This simplifies the process of repeated multiplication.

Understanding exponents is crucial, as they allow us to condense the expression of large numbers. For instance, writing 1,000,000 as \(10^6\) is much neater and clearer. Exponents are used in various mathematical fields, from algebra to calculus and beyond.

• The base is the number being multiplied.
• The exponent indicates how many times to multiply the base by itself.
• Exponents are a shorthand notation for repeated multiplication.

Logarithms

Logarithms act as the inverse operation of exponents. When you encounter an expression like \(b^e = n\), you can rewrite it in logarithmic form as \(\log_b n = e\). The logarithm tells us what power the base needs to be raised to, to produce the result. In our example, \(3^3 = 27\) converts to \(\log_3 27 = 3\). This asks the question: "To what power must 3 be raised to result in 27?"

Logarithms have important applications in science and engineering, where they help solve problems involving exponential growth or decay. They can also help simplify complex calculations of multiplication and division.

• The base of the logarithm is the same as the base of the exponent.
• The result of the logarithm is the same as the exponent in the power expression.
• Logarithms provide a way to "undo" exponentiation, making them vital for solving exponential equations.

Mathematical Notation

Mathematical notation serves as the language of mathematics, providing a clear and concise way to represent mathematical ideas. These symbols and formats allow mathematicians and students to communicate complex concepts precisely.

In our example, we are dealing with both exponential and logarithmic notations. The exponential notation \(3^3 = 27\) enables the expression of repeated multiplication cleanly. When this is rewritten in logarithmic notation \(\log_3 27 = 3\), it offers another perspective on the same mathematical relationship between the numbers.

• Exponents are represented by a superscript on the base number.
• Logarithms use the "log" symbol followed by a subscript base number.
• Notation reduces complex mathematical ideas into simple symbols and formats.

Understanding these notations is essential as they come up often in various areas of math, from solving equations to calculating growth rates, and more.