Problem 32
Question
Write logarithm as a sum. Then simplify, if possible. \(\log _{3}(27 \cdot 5)\)
Step-by-Step Solution
Verified Answer
\( \log_{3}(27 \cdot 5) = 3 + \log_{3}(5) \); cannot be simplified further.
1Step 1: 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: \( \log_{3}(27 \cdot 5) = \log_{3}(27) + \log_{3}(5) \).
2Step 2: 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, \( \log_{3}(3^3) = 3 \cdot \log_{3}(3) = 3 \times 1 = 3 \).
3Step 3: 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.
Key Concepts
Logarithm Product RulePower RuleSimplification of Logarithms
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.
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.
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.
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.
Other exercises in this chapter
Problem 32
Let \(f(x)=2 x-5\) and \(g(x)=x+1 .\) Find each of the following function values. $$ (f / g)(2) $$
View solution Problem 32
Graph each function. $$ f(x)=\frac{1}{2} e^{x} $$
View solution Problem 33
Solve each equation. Give the exact solution and an approximation to four decimal places. See Example 4. $$ 2^{x+1}=3^{x} $$
View solution Problem 33
Let \(s(x)=3-x\) and \(t(x)=x^{2}-x-6 .\) Find each function value. $$ (s \cdot t)(-2) $$
View solution