Problem 24 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{3} 27 $$

Step-by-Step Solution

Answer

The value of the expression \(\log _{3} 27\) is 3.

Step 1: Analyzing the Problem Statement

Here, we have a logarithmic expression \(\log _{3} 27\). This is the base 3 logarithm of 27, and it is equivalent to asking for what exponent we must raise the base (in this case, 3) to get 27.

Step 2: Using Logarithm properties

We know from the definition of logarithms that if \(b^y = x\), then \(\log _{b} x = y\). Thus, if 3 raised to a certain power equals 27, then \(\log _{3} 27\) will equal that power.

Step 3: Applying Logarithm properties to find the exponent

We know that \(3^3 = 27\). Thus, 3 raised to the power 3 yields 27. Therefore, in the expression \(\log _{3} 27\), the power to which we need to raise 3 to obtain 27 is 3. Hence, \(\log _{3} 27 = 3\).

Key Concepts

Logarithmic Expressions

A logarithmic expression is a unique mathematical way of expressing the relationship between numbers through exponents. When you see something like \(\log_{3} 27\), it might seem a bit mysterious at first. However, it's simply asking, "What power do I need to raise 3 to, in order to get 27?" Logarithmic expressions make it easier to work with numbers that are too large or too complex when used directly as exponents.

Logarithms serve an important purpose in solving equations, particularly in balancing equations and modeling real-world scenarios. For instance:

• If \(b^y = x\), then \(\log_{b} x = y\).
• In our example, \(3^y = 27\), and we found that \(y = 3\) because \(3^3 = 27\).

Grasping logarithms opens the door to understanding how exponential growth works, such as in population models or interest calculations.

Base of a Logarithm

The base of a logarithm is a critical component in logarithmic expressions. It tells us what number is being repeatedly multiplied in order to reach a target value. In the expression \(\log_{3} 27\), the base is 3.

The base is what we are working with in terms of powers or exponents. For example:

• If you have \(\log_{b} x = y\), the base \(b\) is the number being raised to the power of \(y\).
• In \(\log_{3} 27\), the base 3 is raised to the power 3 to give us 27.

Bases in logarithms are typically positive numbers greater than 1. The choice of base often depends on the context. For example, base 10 is common in scientific work, while base 2 is prevalent in computing.

Properties of Logarithms

Understanding the properties of logarithms helps simplify and solve many logarithmic expressions. These properties are rules that govern how logarithms operate and can be applied to solve problems more efficiently.

Here are some essential properties of logarithms:

• Product Rule: \(\log_{b}(xy) = \log_{b} x + \log_{b} y\). This rule allows you to break down the logarithm of a product into a sum of logarithms.
• Quotient Rule: \(\log_{b}(\frac{x}{y}) = \log_{b} x - \log_{b} y\). This property simplifies the logarithm of a quotient into a subtraction.
• Power Rule: \(\log_{b}(x^y) = y\log_{b} x\). This rule lets you move the exponent in front of the logarithm as a multiplier.
• Change of Base Formula: This allows converting between bases: \(\log_{b} x = \frac{\log_{k} x}{\log_{k} b}\) where \(k\) is a new base, typically 10 or \(e\) (for natural logs).

These properties enable simplification and solving of complex logarithmic equations. In the example of \(\log_{3} 27\), we applied the power rule in reverse to find that \(3^3 = 27\), simplifying our solution to find \(\log_{3} 27 = 3\).