Problem 24 - College Algebra

Question

Evaluate each expression without using a calculator. $$ \log _{3} 27 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Logarithmic Expression

The logarithmic expression we need to evaluate is \(\log _{3} 27\). This represents the power or exponent to which 3 must be raised to get 27. If we rewrite this as an exponential equation, it becomes \(3^x = 27\).

Step 2: Evaluating the Expression

We need to find a number \(x\) such that \(3^x = 27\). If we try \(x=2\), we get \(3^2 = 9\), which is less than 27. If we try \(x=3\), we get \(3^3 = 27\), which equals 27. Hence, \(x=3\) is the solution.

Key Concepts

Exponential Equations

An exponential equation is a mathematical statement where a number, known as the base, is raised to a variable exponent. Exponential equations often appear when solving logarithmic expressions. For example, finding the value of \(\log _{3} 27\) involves turning the logarithm into an exponential equation: \(3^x = 27\). Here, 3 is the base, and we are solving for the exponent \(x\).
Exponential equations are expressed in the form \(b^x = y\), where \(b\) is the base and \(x\) is the exponent. The goal is to determine \(x\) given \(b\) and \(y\).
To solve them, you can:

• Guess and check: Substitute different numbers for the exponent until the equation holds true.
• Use logarithms: Rewrite the equation in logarithmic form to isolate the exponent.

These methods help in understanding the relationship between exponential and logarithmic expressions.

Evaluating Logarithms

Evaluating logarithms involves finding the exponent to which a base must be raised to achieve a certain number. Consider the logarithm \(\log _{3} 27\). This expression means we are looking for what power 3 needs to be raised to get 27.
To evaluate this logarithm without a calculator, follow these steps:

• Rewrite the logarithmic expression as an exponential equation: \(3^x = 27\).
• Determine the value for \(x\) that satisfies this equation. Through trial and error, you find \(3^3 = 27\), thus \(x = 3\).

By rewriting the logarithmic expression as an exponential equation, you utilize the relationship between exponents and logarithms to find the solution. This method helps in visualizing the operation and reasoning behind the solution.

Base of Logarithm

In logarithms, the base is a fundamental component and is the number that is repeatedly multiplied by itself. In \(\log _{b} y = x\), \(b\) is the base, \(y\) is the result, and \(x\) is the exponent.
When dealing with different logarithmic expressions, it's crucial to understand that the base affects how you interpret and solve the problem.

• A base tells you the number that is used as the repeated factor in an exponential form.
• In our example, \(\log _{3} 27\), the base \(3\) indicates that 3 is the number being exponentiated.

Understanding the base allows you to convert between logarithmic and exponential forms easily.
Knowing that in \(3^x = 27\), we use 3 as the base, is key to solving it as an exponential equation. This understanding bridges the gap between logarithms and exponential functions, ensuring a deeper comprehension of how logarithms work.