Problem 11
Question
Match each expression with an equivalent expression from the list on the right. a. \(\frac{\log 11}{\log 3}\) b. \(11 \log _{3} 10\) c. \(\log _{3} 5+\log _{3} 2\) d. \(\log _{3} 10-\log _{3} 11\) $$ \log _{3} 10 $$
Step-by-Step Solution
Verified Answer
Expression c matches with \( \log_{3} 10 \).
1Step 1: Identify Key Properties of Logarithms
To solve this matching problem, we need to utilize key logarithmic identities like the change of base formula, the product rule, and the quotient rule. This will help us transform each expression to see which ones match with \( \log_{3} 10 \).
2Step 2: Simplify Expression a
Expression a is \( \frac{\log 11}{\log 3} \). Using the change of base formula, \( \log_{3} 11 = \frac{\log 11}{\log 3} \). This expression does not match \( \log_{3} 10 \).
3Step 3: Simplify Expression b
Expression b is \( 11 \log_{3} 10 \), which simplifies to \( \log_{3} (10^{11}) \). This is not equivalent to \( \log_{3} 10 \) since the exponents are different.
4Step 4: Simplify Expression c
Expression c is \( \log_{3} 5 + \log_{3} 2 \). By using the product rule, we can combine this to form \( \log_{3} (5 \times 2) = \log_{3} 10 \). This matches \( \log_{3} 10 \).
5Step 5: Simplify Expression d
Expression d is \( \log_{3} 10 - \log_{3} 11 \). By using the quotient rule, we get \( \log_{3} \left( \frac{10}{11} \right) \). This does not match \( \log_{3} 10 \).
6Step 6: Conclusion: Identify Matching Expression
The only expression that matches \( \log_{3} 10 \) directly after simplification is expression c, \( \log_{3} 5 + \log_{3} 2 \), which simplifies to \( \log_{3} 10 \).
Key Concepts
Change of Base FormulaProduct RuleQuotient Rule
Change of Base Formula
The change of base formula is a useful tool when dealing with logarithms. It allows us to convert a logarithm from one base to another. This is particularly helpful when we want to solve logarithms using a calculator, as most calculators only support base 10 (common logarithms) or base e (natural logarithms).
The formula is given by:
In this approach, we can effectively transform any logarithm to a form compatible with our calculator inputs, by choosing \( c \) as 10 or e.
For example, the expression from the exercise:
This enables us to compare or compute values that aren't otherwise straightforward, ensuring more flexibility in handling logarithmic expressions.
The formula is given by:
- \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \)
In this approach, we can effectively transform any logarithm to a form compatible with our calculator inputs, by choosing \( c \) as 10 or e.
For example, the expression from the exercise:
- \( \frac{\log 11}{\log 3} \)
This enables us to compare or compute values that aren't otherwise straightforward, ensuring more flexibility in handling logarithmic expressions.
Product Rule
The product rule for logarithms is helpful when we need to consolidate multiple logarithms with the same base. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors.
In mathematical terms:
This rule allows us to simplify expressions by combining logarithms.
In the exercise, we have the expression:
Remember, the product rule only applies when the bases of the logs are the same. Mixing bases in logarithmic operations requires different strategies for simplification.
In mathematical terms:
- \( \log_{b}(xy) = \log_{b}x + \log_{b}y \)
This rule allows us to simplify expressions by combining logarithms.
In the exercise, we have the expression:
- \( \log_{3} 5 + \log_{3} 2 \)
- \( \log_{3}(5 \times 2) = \log_{3}10 \)
Remember, the product rule only applies when the bases of the logs are the same. Mixing bases in logarithmic operations requires different strategies for simplification.
Quotient Rule
The quotient rule is another important logarithmic identity, especially useful when dealing with divisive expressions of logarithms. According to the quotient rule, the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.
Matted in a formula, the quotient rule is:
This rule enables us to break down division into subtraction, making logarithmic operations easier to handle.
In our given exercise, the expression:
Understanding this rule not only helps in solving mathematical problems but also solidifies your comprehension of logarithmic operations in general.
Matted in a formula, the quotient rule is:
- \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y \)
This rule enables us to break down division into subtraction, making logarithmic operations easier to handle.
In our given exercise, the expression:
- \( \log_{3} 10 - \log_{3} 11 \)
- \( \log_{3}\left(\frac{10}{11}\right) \)
Understanding this rule not only helps in solving mathematical problems but also solidifies your comprehension of logarithmic operations in general.
Other exercises in this chapter
Problem 11
Complete each solution. Let \(f(x)=3 x-1\) and \(g(x)=2 x+3 .\) Find \(f \cdot g\). $$ \begin{aligned} (f \cdot g)(x) &=f(x) \cdot \square \\ &\left.=\square (2
View solution Problem 11
To find \(\ln e^{2},\) we ask, "To what power must we raise ____ to get \(e^{2} ? "\) since the answer is the 2 nd power, \(\ln e^{2}=\) _____.
View solution Problem 11
Fill in the blanks. Perform a check to determine whether 4 is a solution of \(\log _{5}(x+1)=2\)
View solution Problem 11
Fill in the blanks. \(\log _{b} x\) is the _____ to which \(b\) is raised to get \(x.\)
View solution