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Questions and Answers
Which of the following is a valid application of the product rule of logarithms?
Which of the following is a valid application of the product rule of logarithms?
- $\log_b(M / N) = \log_b(M) + \log_b(N)$
- $\log_b(M - N) = \log_b(M) - \log_b(N)$
- $\log_b(M \cdot N) = \log_b(M) + \log_b(N)$ (correct)
- $\log_b(M + N) = \log_b(M) \cdot \log_b(N)$
According to the quotient rule of logarithms, how can $\log_b(\frac{M}{N})$ be rewritten?
According to the quotient rule of logarithms, how can $\log_b(\frac{M}{N})$ be rewritten?
- $\log_b(M) - \log_b(N)$ (correct)
- \(\frac{\log_b(M)}{\log_b(N)}\)
- $\log_b(M) + \log_b(N)$
- $\log_b(M) \cdot \log_b(N)$
If $\log_b(M) = x$ and $\log_b(N) = y$, what is $\log_b(\frac{M}{N})$?
If $\log_b(M) = x$ and $\log_b(N) = y$, what is $\log_b(\frac{M}{N})$?
- x + y
- x - y (correct)
- xy
- \(\frac{x}{y}\)
Given the equation $\log_4(x + 12) + \log_4(x) = 3$, what is the next step after applying the addition law of logarithms?
Given the equation $\log_4(x + 12) + \log_4(x) = 3$, what is the next step after applying the addition law of logarithms?
What is the value of $\log_b(1)$ for any valid base b?
What is the value of $\log_b(1)$ for any valid base b?
What is the simplified form of $log_b(b^k)$?
What is the simplified form of $log_b(b^k)$?
Given the equation $\ln(3x - 2) + \ln(4) = \ln(x + 3)$, what is the result after applying the addition law of logarithms on the left-hand side?
Given the equation $\ln(3x - 2) + \ln(4) = \ln(x + 3)$, what is the result after applying the addition law of logarithms on the left-hand side?
In the equation $\log_4(x+4) - \log_4(x) = 3$, what is the next step after applying the subtraction rule of logarithms?
In the equation $\log_4(x+4) - \log_4(x) = 3$, what is the next step after applying the subtraction rule of logarithms?
After applying the quotient rule to the equation $\log_4(x + 4) - \log_4(x) = 3$ and converting to exponential form, which equation results?
After applying the quotient rule to the equation $\log_4(x + 4) - \log_4(x) = 3$ and converting to exponential form, which equation results?
What condition must be met for M and N in the logarithmic expressions $\log_b(M)$ and $\log_b(N)$?
What condition must be met for M and N in the logarithmic expressions $\log_b(M)$ and $\log_b(N)$?
What is the value of b in the expression $\log_b(b) = $?
What is the value of b in the expression $\log_b(b) = $?
Applying the power rule to the expression $\log_b(M^k)$, the result is:
Applying the power rule to the expression $\log_b(M^k)$, the result is:
If you have the equation $2 \cdot \log_4(x + 4) = 3$, what is the first step to solve for x?
If you have the equation $2 \cdot \log_4(x + 4) = 3$, what is the first step to solve for x?
After applying the power rule of logarithms to the equation $4 \ln(5x - 2) = \ln(x^2 + 16)$, what is the resulting equation?
After applying the power rule of logarithms to the equation $4 \ln(5x - 2) = \ln(x^2 + 16)$, what is the resulting equation?
Using logarithm properties, determine if $M = N$ if $\log_b(M) = \log_b(N)$.
Using logarithm properties, determine if $M = N$ if $\log_b(M) = \log_b(N)$.
Given the equation $\log_b (M \cdot N) = \log_b M + \log_b N$, what does this equation represent?
Given the equation $\log_b (M \cdot N) = \log_b M + \log_b N$, what does this equation represent?
For what values of b is the logarithmic function $\log_b(x)$ defined?
For what values of b is the logarithmic function $\log_b(x)$ defined?
If $\log_b(k) = x$, then what is $b^x$ equal to?
If $\log_b(k) = x$, then what is $b^x$ equal to?
When solving logarithmic equations, why is it important to check the resulting values?
When solving logarithmic equations, why is it important to check the resulting values?
What is the purpose of applying logarithmic rules when solving logarithmic equations?
What is the purpose of applying logarithmic rules when solving logarithmic equations?
Flashcards
Product Rule of Logarithms
Product Rule of Logarithms
The logarithm of a product is the sum of the logarithms of the factors: logb(M*N) = logb(M) + logb(N).
Quotient Rule of Logarithms
Quotient Rule of Logarithms
The logarithm of a quotient is the difference of the logarithms: logb(M/N) = logb(M) - logb(N).
Power Rule of Logarithms
Power Rule of Logarithms
The logarithm of a number raised to a power is the product of the power and the logarithm of the number: logb(M^k) = k * logb(M)
Logarithm of One
Logarithm of One
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Logarithm of Base
Logarithm of Base
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Inverse Property of Logarithms
Inverse Property of Logarithms
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Study Notes
- Logarithmic functions are used to solve logarithmic equations to better understand their applications in real life.
- The basic laws concerning logarithms should be understood first.
Laws of Logarithms
- Product Rule: logb(M*N) = logbM + logbN
- Quotient Rule: logb(M/N) = logbM - logbN
- Power Rule: logb(M^k) = k*logbM
- logb(1) = 0
- logb(b) = 1
- logb(b^k) = k
- b^(logb(k)) = k
bmust be greater than 0 and not equal to 1.MandNmust be positive real numbers.
First Law: Addition or Product Rule
- The logarithm of a product is the sum of the logarithms of the factors: logb(M*N) = logbM + logbN
Second Law: Subtraction or Quotient Rule
- The logarithm of the ratio of two quantities is the logarithm of the numerator minus the logarithm of the denominator: logb(M/N) = logbM - logbN
Third Law: Power Rule
- The logarithm of an exponential number is the exponent times the logarithm of the base: logb(M^k) = k * logbM
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