Problem 83 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{3} \frac{1}{27} $$

Step-by-Step Solution

Answer

-3

Step 1: Understanding the Problem

First, one has to understand what a logarithm is. Given a logarithm \( \log _{b} a = n \), this translates to \( b^{n} = a \). Here, \( b \) is the base, \( n \) is the exponent and \( a \) is the number we get when we raise the base to the exponent.

Step 2: Observing the Given Logarithm

In this case, our logarithm is \( \log _{3} \frac{1}{27} \). This means that we are looking for the exponent to which we must raise 3 to get the number 1/27.

Step 3: Finding the Value

We know that \( 3^{-3} = \frac{1}{27} \). Thus, -3 is the exponent which we must raise the base 3 to get the number 1/27. Consequently, \( \log _{3} \frac{1}{27} = -3 \). Therefore, the answer is -3.

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of mathematics that describe situations where a quantity grows or decays at a constant rate. These functions are usually represented as \( y = b^x \), where \( b \) is the base and \( x \) is the exponent. The base \( b \) is a constant that dictates the rate of growth or decay, while the exponent \( x \) varies to show different values of the function.

One of the most important features of exponential functions is their consistent rate of increase or decrease. If \( b > 1 \), the function experiences exponential growth; if \( 0 < b < 1 \), the function experiences exponential decay. For example:

• Growth: \( y = 2^x \) means the value doubles for each increase in \( x \).
• Decay: \( y = \left(\frac{1}{2}\right)^x \) means the value halves for each increase in \( x \).

Outside of pure mathematics, exponential functions are used to model real-world phenomena such as population growth, radioactive decay, and compound interest. This makes them particularly important to understand, as they provide insights into processes that change rapidly over time.

Exponents

Exponents are a way to express repeated multiplication of the same number. When we write \( b^n \), \( b \) is called the base and \( n \) is the exponent, indicating that \( b \) should be multiplied by itself \( n \) times. For instance, \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \).

Understanding exponents is crucial for working with powers of numbers. There are several key rules involving exponents that make calculations easier:

• Product of Powers Rule: \( b^m \times b^n = b^{m+n} \).
• Quotient of Powers Rule: \( \frac{b^m}{b^n} = b^{m-n} \).
• Power of a Power Rule: \((b^m)^n = b^{m\cdot n} \).
• Negative Exponent Rule: \( b^{-n} = \frac{1}{b^n} \).
• Zero Exponent Rule: \( b^0 = 1 \) for any \( b eq 0 \).

These rules allow us to simplify expressions and solve equations that involve powers, significantly streamlining processes in algebra and calculus.

Base of a Logarithm

The base of a logarithm is a number that helps define the "scaling" factor in logarithmic expressions. In a logarithmic expression such as \( \log_b a \), \( b \) is the base. The statement \( \log_b a = n \) is interpreted as saying that \( b^n = a \). This means that when you take the logarithm of a number, you are finding the power to which the base must be raised to produce that number.

The base is critical as it determines the behavior and properties of the logarithmic function. Common bases include:

• Base 10 (common logarithms), often written as \( \log a \).
• Base \( e \) (natural logarithms), written as \( \ln a \), where \( e \approx 2.718 \).
• Base 2 (binary logarithms), frequently used in computer science.

Changing the base can change the output of the logarithm, but the relationship between different logarithmic bases can be found using the change of base formula: \( \log_b a = \frac{\log_k a}{\log_k b} \). This makes it possible to convert between logarithms of different bases, offering flexibility in various mathematical and practical contexts.