Problem 35 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{3} 729 $$

Step-by-Step Solution

Answer

\( \log_{3} 729 = 6 \).

Step 1: Understand the Problem

We need to evaluate the logarithmic expression \( \log_{3} 729 \). This expression means we are looking for the power to which 3 must be raised to obtain 729.

Step 2: Express 729 as a Power of 3

To solve \( \log_{3} 729 \), we first need to express 729 as a power of 3. We start by dividing 729 by 3 repeatedly to see how many times 3 divides into 729.1. \( 729 \div 3 = 243 \)2. \( 243 \div 3 = 81 \)3. \( 81 \div 3 = 27 \)4. \( 27 \div 3 = 9 \)5. \( 9 \div 3 = 3 \)6. \( 3 \div 3 = 1 \)Counting each division gives us six 3s, meaning \( 729 = 3^6 \).

Step 3: Evaluate the Logarithm

Since \( 729 = 3^6 \), it follows that \( \log_{3} 729 = \log_{3} (3^6) \). Using the logarithmic identity \( \log_{b} (b^n) = n \), we can simplify:\[ \log_{3} (3^6) = 6 \].

Step 4: Final Answer

We have determined that \( \log_{3} 729 = 6 \). This means 3 must be raised to the power of 6 to yield 729.

Key Concepts

Evaluating Logarithms

When you evaluate a logarithm, you're essentially asking a question: "To what power must the base be raised to produce a given number?" In our example, we have the expression \( \log_{3} 729 \), which asks, "3 raised to what power gives us 729?"
To answer this, we express the given number, 729, as a power of the base, which is 3 in this case. If you find that 729 equals \( 3^6 \), then your answer is 6, because that's the power we need for 3 to reach 729.
This process involves breaking down the problem step by step:

• Identify the base of the logarithm (here, it's 3).
• Express the number you're taking the logarithm of (729) as a power of this base.
• Recognize that the exponent is your answer.

Evaluating logarithms is useful because it helps us transform complex numbers into more manageable forms by understanding them through their relationships with their bases.

Powers of Numbers

The concept of powers is crucial, especially when dealing with logarithms. A power of a number involves multiplying that number by itself a certain number of times. In the original problem, we expressed 729 as \( 3^6 \).
Let's break down why this happened:

• It's all about repeated multiplication: \(3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729\).
• This can be written using exponent notation as \(3^6\), indicating six repetitions of multiplication.

Understanding how to convert a number into its power form helps in evaluating logarithms because it allows you to work directly with exponents. Powers simplify multiplication into a clearer format where a base number is multiplied by itself several times.

Logarithmic Identities

Logarithmic identities are rules or properties that simplify the handling of logarithms. They provide shortcuts and simplify expression evaluation. One essential identity used in the exercise is \(\log_{b} (b^n) = n\). This identity states that if you take the logarithm of a number that is the base raised to a power, the result is the exponent.
Here's why this identity is fundamental:

• It connects exponentiation and logarithms directly, making calculations quicker and more intuitive.
• If you have \(b^n\) as a number, the logarithm with the same base "undoes" the power, leaving you with the exponent \(n\).

In our example, using this identity directly led us to evaluate \(\log_{3} 729\) as 6 because we expressed 729 as \(3^6\). Knowing logarithmic identities allows you to work with these expressions efficiently. It’s like having a toolkit for unlocking complex mathematical problems.