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Chapter 11: Problem 32

Write logarithm as a sum. Then simplify, if possible. \(\log _{3}(27 \cdot 5)\)

Short Answer

Expert verified
\( \log_{3}(27 \cdot 5) = 3 + \log_{3}(5) \); cannot be simplified further.

Step by step solution

01

Apply Logarithm Product Rule

The logarithm of a product can be expressed as the sum of the logarithms of the individual factors. For the expression \( \log_{3}(27 \cdot 5) \), use the product rule: \( \log_{3}(27 \cdot 5) = \log_{3}(27) + \log_{3}(5) \).
02

Simplify the First Logarithm

Find \( \log_{3}(27) \). Since \( 27 \) is equal to \( 3^3 \), this simplifies to \( \log_{3}(3^3) \). Using the power rule, \( \log_{3}(3^3) = 3 \cdot \log_{3}(3) = 3 \times 1 = 3 \).
03

Combine the Simplified Logarithms

Substitute \( \log_{3}(27) = 3 \) back into the expression. Thus, \( \log_{3}(27 \cdot 5) = \log_{3}(27) + \log_{3}(5) = 3 + \log_{3}(5) \). Since \( \log_{3}(5) \) cannot be simplified further without a calculator, this is the final simplified expression for the logarithm as a sum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Product Rule
Logarithms have special rules that make dealing with products, quotients, and powers easier. One of these rules is the **product rule**. This rule says that the logarithm of a product is the same as the sum of the logarithms of the individual terms. If you have \(a \cdot b\), then \(\log_{b}(a \cdot b) = \log_{b}(a) + \log_{b}(b)\).
Let's take a look at how this works with the original problem: \(\log_{3}(27 \cdot 5)\). Using the product rule, you can split the logarithm as follows:
\(\log_{3}(27 \cdot 5) = \log_{3}(27) + \log_{3}(5)\). This step simplifies complex expressions, breaking them into manageable parts.
Power Rule
The **power rule** for logarithms helps you deal with expressions like \(\log_{b}(a^n)\). This rule states that you can bring the exponent \(n\) in front of the log, resulting in \(n \cdot \log_{b}(a)\). It transforms power forms into a simpler multiplication.
Consider the expression \(\log_{3}(27)\). Since \(27\) can be expressed as \(3^3\), this becomes \(\log_{3}(3^3)\).
According to the power rule, this simplifies to \(3 \cdot \log_{3}(3)\). Since \(\log_{3}(3) = 1\), it further reduces to \(3 \times 1 = 3\). By recognizing the exponent, the power rule makes the problem much less intimidating.
Simplification of Logarithms
Simplification is the key to making logarithms manageable and easier to understand. After applying the product and power rules, you are often left with a simpler expression. This exercise led to breaking down \(\log_{3}(27 \cdot 5)\) into \(3 + \log_{3}(5)\).
This is as simplified as you can get without a calculator because \(\log_{3}(5)\) is not an integer and can't be further reduced. Always look for ways to apply the product, quotient, or power rules first; they are powerful tools for simplification.
These simplifications not only assist in solving problems but also help in gaining a deeper understanding of logarithms.

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Most popular questions from this chapter

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log (x+90)+\log x=3 $$

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