Problem 58 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{3} \frac{1}{9}=-2 $$

Step-by-Step Solution

Answer

The exponential form of the equation is \(3^{-2} = 1/9\).

Step 1: Identify the elements

The base is 3, the result inside the logarithm is 1/9, and the value of the logarithm is -2.

Step 2: Write in exponential form

Write these values in the form 'base raised to the exponent equals the result'. This gives 3 raised to the power of -2 equals 1/9.

Key Concepts

Logarithmic Equations

Logarithmic equations allow us to connect exponents with their results in a different and often more straightforward way. A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, in the original exercise \(\log _{3} \frac{1}{9}=-2\), the logarithm is telling us that if you raise the base 3 to the power of -2, you will get \(\frac{1}{9}\).

Logarithmic equations are very helpful in solving problems involving exponentials because they provide a clear way to reverse or "undo" exponentiation. When you see a logarithmic equation, you can rewrite it in exponential form to understand it more deeply.

To convert a logarithmic equation into exponential form:

• Identify the "base" of the logarithm.
• Recognize the "result" inside the log operation.
• Translate: \(\log_{b} a = c\) becomes \(b^{c} = a\).

Exponents

Exponents are shorthand notation for repeated multiplication of a base number. They indicate how many times the base is multiplied by itself. For example, \(3^2\) means that the base 3 is multiplied by itself twice, giving a result of 9. This concept scales to different types of exponents, such as negative exponents.

Negative exponents signify division rather than multiplication. They tell us that the base is on the bottom of a fraction. Thus, \(3^{-2}\) can be translated to \(\frac{1}{3^2}\), which simplifies to \(\frac{1}{9}\).

When dealing with exponents, it’s important to understand these rules:

• Multiplication leads to positive exponents.
• Division leads to negative exponents.
• An exponent of zero makes the base equal 1, i.e., \(b^0 = 1\).

Recognizing exponential form from logarithmic equations helps simplify and solve for unknown quantities.

Base of a Logarithm

The base in a logarithm is essentially the number which is raised to an unknown power to get another number. It's a central aspect of understanding logarithms and their conversion to exponential form. In the equation \(\log_{3} \frac{1}{9}=-2\), 3 is the base.

The base tells us what number we are repeatedly multiplying. In our exercise, since the base is 3, we ask: "To what power should 3 be raised to obtain \(\frac{1}{9}\)?"

Different bases describe different logarithm systems:

• Base 10 is common in scientific and daily use, often written as \(\log\).
• Base \(e\) (approximately 2.718) is used in natural logarithms, noted as \(\ln\).
• Any other base is usually specified, as in this exercise with base 3.

Understanding the base helps interpret and correctly convert logarithmic expressions into exponential ones, allowing us to solve more complex mathematical problems.