Questions and Answers
Which of the following statements is true?
In the definition of the exponential function f(x)=b^x, what are the stipulations for the base b?
Which of the following statements is not true for the graph f(x)=b^x?
Which of the following is not true about the expression log7 49?
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To what is the expression logb b^x equal?
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Which of the following statements is not true for the graph of y=logb x for b>1?
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The domain of f(x)=logb[g(x)] can be determined by finding the solution to which inequality?
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Which of the following statements is not a property of logarithms?
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Which of the following is not true?
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Which of the following is true?
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Why is the logarithmic property of equality, which says that 'if logb u=logb v, then u=v' true?
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If a and b are positive real numbers such that a≠1, b≠1, and u is any positive real number, then the logarithmic expression logb u is equivalent to which of the following?
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Logarithms are studied for which of the following reasons?
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In solving the exponential equation 4^x=7^(x-1), which of the following is not true?
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Study Notes
Rational Inequalities
- Boundary points determined by the denominator set to zero should be represented with an open circle on a number line.
- Boundary points determined by the numerator set to zero may not necessarily represent an open circle.
Exponential Functions
- The base ( b ) of the exponential function ( f(x)=b^x ) must be greater than zero and not equal to one.
- As ( x ) approaches negative infinity, the graph of ( f(x)=b^x ) approaches zero.
Logarithmic Properties
- The expression ( \log_7 49 ) represents the power that 7 must be raised to in order to yield 49, hence it equals 2.
- The term ( \log_b b^x ) simplifies to ( x ) for ( b>0 ) and ( b \neq 1 ).
Graphs of Logarithmic Functions
- The graph of ( y=\log_b x ) passes through the point (1,0) and (b,1), with a vertical asymptote at ( x=0 ).
- For ( b>1 ), the graph of ( y=\log_b x ) is decreasing on the interval ( (0, \infty) ) is false; it actually increases.
Domain Restrictions
- The domain of ( f(x)=\log_b[g(x)] ) is determined by the inequality ( g(x)>0 ).
Properties of Logarithms
- Logarithmic identities include:
- ( \log_b \frac{u}{v} = \log_b u - \log_b v )
- ( \log_b u^r = r \log_b u )
- ( \log_b 1 = 0 )
- An important logarithmic identity, ( \ln e^8 = 8 ), must hold true.
Logarithmic Function Uniqueness
- The property that if ( \log_b u = \log_b v ), then ( u = v ) is valid due to the one-to-one nature of logarithmic functions.
Change of Base Formula
- The logarithmic expression ( \log_b u ) can be transformed using the change of base formula:
- ( \log_b u = \frac{\log_a u}{\log_a b} )
Use of Logarithms in Equations
- Logarithms simplify the solving of exponential equations when direct base comparisons are not viable, assisting in more complex calculations.
Solving Exponential Equations
- In the equation ( 4^x = 7^{(x - 1)} ), transforming it involves:
- ( x \ln 4 = (x - 1) \ln 7 )
- From this relation, if ( x\ln \frac{4}{7} = -\ln 7 ), it leads to the solution ( x = -0.2005 ).
Studying That Suits You
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Description
Test your knowledge of rational inequalities with these flashcards. Focus on the critical concept of boundary points and how to represent them on a number line. This quiz is perfect for students using MyMathLab to strengthen their understanding.
