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Magazine Chapter 6: Problem 5
Use rules of logarithms to find the value of \(x\). Verify your answer with a calculator. a. \(\ln x=\ln 2+\ln 5\) b. \(\ln x=\ln 24-\ln 2\) c. \(\ln x^{2}=2 \ln 11\) d. \(\ln x=3 \ln 2+2 \ln 6\) e. \(\ln x=6 \ln 2-2 \ln 3\) f. \(\ln x=4 \ln 2-3 \ln 2\)
Short Answer
Expert verified
a) x = 10, b) x = 12, c) x = 11, d) x = 288, e) x = 64/9, f) x = 2.
Step by step solution
01
Problem a: Combine Logarithms Using Product Rule
Given: \( \ln x = \ln 2 + \ln 5 \) \Use the product rule of logarithms: \( \ln a + \ln b = \ln (a \cdot b) \) \Thus: \( \ln x = \ln (2 \cdot 5) \) \=> \( \ln x = \ln 10 \) \Therefore, \( x = 10 \).
02
Problem b: Combine Logarithms Using Quotient Rule
Given: \( \ln x = \ln 24 - \ln 2 \) \Use the quotient rule of logarithms: \( \ln a - \ln b = \ln (a \div b) \) \Thus: \( \ln x = \ln (24 \div 2) \) \=> \( \ln x = \ln 12 \) \Therefore, \( x = 12 \).
03
Problem c: Simplify Using Power Rule
Given: \( \ln x^{2} = 2 \ln 11 \) \Use the power rule of logarithms: \( \ln a^{b} = b \ln a \) \Thus: \( \ln (x^{2}) = 2 \ln 11 \) \Since \( \ln (x^{2}) = 2 \ln x \), we have: \( 2 \ln x = 2 \ln 11 \) \=> \( \ln x = \ln 11 \) \Therefore, \( x = 11 \).
04
Problem d: Combine Logarithms Using Product Rule
Given: \( \ln x = 3 \ln 2 + 2 \ln 6 \) \Rewrite using the power rule: \( 3 \ln 2 = \ln 2^{3} \), and \( 2 \ln 6 = \ln 6^{2} \) \Thus: \( \ln x = \ln 2^{3} + \ln 6^{2} \) \Use the product rule of logarithms: \( \ln a + \ln b = \ln (a \cdot b) \) \=> \( \ln x = \ln (2^{3} \cdot 6^{2}) \) \=> \( \ln x = \ln (8 \cdot 36) \) \=> \( \ln x = \ln 288 \) \Therefore, \( x = 288 \).
05
Problem e: Combine Logarithms Using Quotient Rule
Given: \( \ln x = 6 \ln 2 - 2 \ln 3 \) \Rewrite using the power rule: \( 6 \ln 2 = \ln 2^{6} \), and \( 2 \ln 3 = \ln 3^{2} \) \Use the quotient rule of logarithms: \( \ln a - \ln b = \ln (a \div b) \) \Thus: \( \ln x = \ln 2^{6} - \ln 3^{2} \) \=> \( \ln x = \ln (2^{6} \div 3^{2}) \) \=> \( \ln x = \ln (64 \div 9) \) \=> \( \ln x = \ln (64/9) \) \Therefore, \( x = 64/9 \).
06
Problem f: Simplify Using Power Rule
Given: \( \ln x = 4 \ln 2 - 3 \ln 2 \) \Combine the terms: \( 4 \ln 2 - 3 \ln 2 = (4 - 3) \ln 2 \) \Thus: \( \ln x = 1 \ln 2 \) \=> \( \ln x = \ln 2 \) \Therefore, \( x = 2 \).
07
Verification with Calculator
Verify each solved value against a calculator to ensure the correctness. E.g., for part a, check \( \ln 10 = \ln 2 + \ln 5 \). Perform similar verifications for all parts.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
product rule of logarithms
The product rule of logarithms is a key concept when dealing with logarithmic expressions. It's very helpful in combining multiple logarithms into a single one. The rule states that for any positive numbers \( a \) and \( b \): \(\ln a + \ln b = \ln (a \cdot b)\).
This means, adding two log values is equivalent to the log of their products.
For example, in Problem a, we have \( \ln x = \ln 2 + \ln 5 \).
Using the product rule: \( \ln x = \ln (2 \cdot 5) \), which simplifies to \( \ln x = \ln 10 \). Hence, \( x = 10 \).
Try practicing more examples with different values to fully understand and reinforce this concept.
This means, adding two log values is equivalent to the log of their products.
For example, in Problem a, we have \( \ln x = \ln 2 + \ln 5 \).
Using the product rule: \( \ln x = \ln (2 \cdot 5) \), which simplifies to \( \ln x = \ln 10 \). Hence, \( x = 10 \).
Try practicing more examples with different values to fully understand and reinforce this concept.
quotient rule of logarithms
The quotient rule of logarithms is another fundamental concept. It allows us to condense logarithms by expressing them as the logarithm of a quotient. The rule states that for any positive numbers \( a \) and \( b \): \( \ln a - \ln b = \ln (a / b) \).
In simpler terms, subtracting one log from another equals the log of their division.
For instance, in Problem b, given \( \ln x = \ln 24 - \ln 2 \), we can use the quotient rule: \( \ln x = \ln (24 / 2) \).
This simplifies to \( \ln x = \ln 12 \), and hence, \( x = 12 \).
Practice with different values to become more comfortable with this rule.
In simpler terms, subtracting one log from another equals the log of their division.
For instance, in Problem b, given \( \ln x = \ln 24 - \ln 2 \), we can use the quotient rule: \( \ln x = \ln (24 / 2) \).
This simplifies to \( \ln x = \ln 12 \), and hence, \( x = 12 \).
Practice with different values to become more comfortable with this rule.
power rule of logarithms
The power rule of logarithms is vital when exponents are involved. It helps simplify expressions where the logarithm has an exponent. The rule states that for any positive number \( a \) and any real number \( b \): \( \ln (a^b) = b \ln a \).
This means the exponent can be moved in front as a coefficient.
For example, in Problem c, we are given \( \ln x^{2} = 2 \ln 11 \).
Using the power rule: \( \ln (x^{2}) = 2 \ln 11 \), and since \( \ln (x^{2}) = 2 \ln x \), we have \( 2 \ln x = 2 \ln 11 \).
Simplifying, we get \( \ln x = \ln 11 \), and hence, \( x = 11 \).
This rule is very handy for dealing with exponents within logarithms.
This means the exponent can be moved in front as a coefficient.
For example, in Problem c, we are given \( \ln x^{2} = 2 \ln 11 \).
Using the power rule: \( \ln (x^{2}) = 2 \ln 11 \), and since \( \ln (x^{2}) = 2 \ln x \), we have \( 2 \ln x = 2 \ln 11 \).
Simplifying, we get \( \ln x = \ln 11 \), and hence, \( x = 11 \).
This rule is very handy for dealing with exponents within logarithms.
solving logarithmic equations
Solving logarithmic equations involves using logarithm properties to isolate the variable. Let's walk through Problem e as an example.
The given equation is: \( \ln x = 6 \ln 2 - 2 \ln 3 \).
First, apply the power rule: \( 6 \ln 2 = \ln 2^{6} \) and \( 2 \ln 3 = \ln 3^{2} \).
Then, use the quotient rule: \( \ln x = \ln (2^{6} / 3^{2}) \). This simplifies to: \( \ln x = \ln (64 / 9) \).
Therefore, \( x = 64 / 9 \).
Practice moving between these rules until the process becomes second nature.
The given equation is: \( \ln x = 6 \ln 2 - 2 \ln 3 \).
First, apply the power rule: \( 6 \ln 2 = \ln 2^{6} \) and \( 2 \ln 3 = \ln 3^{2} \).
Then, use the quotient rule: \( \ln x = \ln (2^{6} / 3^{2}) \). This simplifies to: \( \ln x = \ln (64 / 9) \).
Therefore, \( x = 64 / 9 \).
Practice moving between these rules until the process becomes second nature.
logarithm verification with calculator
Verifying your solutions with a calculator is an important step to ensure accuracy. First, solve the logarithmic equation using the appropriate rules. After finding the value of \( x \), you can verify it with a calculator by comparing the left-hand side and right-hand side of the original equation.
For example, take Problem a: \( \ln 10 = \ln 2 + \ln 5 \).
Use a calculator to compute each term: \( \ln 10 \approx 2.30258 \) and \( \ln 2 + \ln 5 \approx 2.30258 \).
They match, confirming the answer is correct.
Repeat similar verification for each problem to build confidence in your solutions.
For example, take Problem a: \( \ln 10 = \ln 2 + \ln 5 \).
Use a calculator to compute each term: \( \ln 10 \approx 2.30258 \) and \( \ln 2 + \ln 5 \approx 2.30258 \).
They match, confirming the answer is correct.
Repeat similar verification for each problem to build confidence in your solutions.
