Podcast
Questions and Answers
Which of the following statements is true?
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?
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 statements is not true for the graph f(x)=b^x?
Which of the following is not true about the expression log7 49?
Which of the following is not true about the expression log7 49?
Signup and view all the answers
To what is the expression logb b^x equal?
To what is the expression logb b^x equal?
Signup and view all the answers
Which of the following statements is not true for the graph of y=logb x for b>1?
Which of the following statements is not true for the graph of y=logb x for b>1?
Signup and view all the answers
The domain of f(x)=logb[g(x)] can be determined by finding the solution to which inequality?
The domain of f(x)=logb[g(x)] can be determined by finding the solution to which inequality?
Signup and view all the answers
Which of the following statements is not a property of logarithms?
Which of the following statements is not a property of logarithms?
Signup and view all the answers
Which of the following is not true?
Which of the following is not true?
Signup and view all the answers
Which of the following is true?
Which of the following is true?
Signup and view all the answers
Why is the logarithmic property of equality, which says that 'if logb u=logb v, then u=v' true?
Why is the logarithmic property of equality, which says that 'if logb u=logb v, then u=v' true?
Signup and view all the answers
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?
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?
Signup and view all the answers
Logarithms are studied for which of the following reasons?
Logarithms are studied for which of the following reasons?
Signup and view all the answers
In solving the exponential equation 4^x=7^(x-1), which of the following is not true?
In solving the exponential equation 4^x=7^(x-1), which of the following is not true?
Signup and view all the answers
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
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.
