Podcast
Questions and Answers
Which of the following types of functions is a transcendental function? (Select all that apply)
Which of the following types of functions is a transcendental function? (Select all that apply)
- radical
- polynomial
- rational
- logarithmic (correct)
Which of the following functions is not a one-to-one function? (Select all that apply)
Which of the following functions is not a one-to-one function? (Select all that apply)
- f(x) = x³
- ƒ(x) = x² (correct)
- ƒ(x) = x - 4
- ƒ(x) = x
What is the value for the function ƒ(x) = 3^x⁻¹ if x = 4?
What is the value for the function ƒ(x) = 3^x⁻¹ if x = 4?
27
What is the value for the function f(x) = 2e^x if x = 4 (rounded to the hundredths place)?
What is the value for the function f(x) = 2e^x if x = 4 (rounded to the hundredths place)?
Which of the following transformations is represented by 2 in the following equation: f(x) = 3^x⁻² + 5?
Which of the following transformations is represented by 2 in the following equation: f(x) = 3^x⁻² + 5?
Which of the following logarithmic properties is used in the following expansion: log x³ = 3 log x?
Which of the following logarithmic properties is used in the following expansion: log x³ = 3 log x?
What is the argument in the logarithmic function f(x) = log₂4?
What is the argument in the logarithmic function f(x) = log₂4?
To which line are inverse functions symmetric?
To which line are inverse functions symmetric?
Rewrite log₆7,776 = 5 as an exponential equation.
Rewrite log₆7,776 = 5 as an exponential equation.
What type of function is the inverse of a logarithmic function?
What type of function is the inverse of a logarithmic function?
Expand the expression log(xy/z).
Expand the expression log(xy/z).
Expand the expression ln(x²y³).
Expand the expression ln(x²y³).
Condense the following expression to a single logarithm: 3 log x - 4 log y.
Condense the following expression to a single logarithm: 3 log x - 4 log y.
Condense the following expression to a single logarithm: ½ln x + ln y.
Condense the following expression to a single logarithm: ½ln x + ln y.
Evaluate logarithm using the change of base formula: log₅11.
Evaluate logarithm using the change of base formula: log₅11.
Evaluate logarithm using the change of base formula: ln₄200.
Evaluate logarithm using the change of base formula: ln₄200.
Find the inverse function: f(x) = x³ + 9.
Find the inverse function: f(x) = x³ + 9.
Find the inverse function: f(x) = 2x + 5/x.
Find the inverse function: f(x) = 2x + 5/x.
Solve without a calculator: 8^(x²+4) = 8¹³.
Solve without a calculator: 8^(x²+4) = 8¹³.
Solve without a calculator: x = log₈4 + log₈16.
Solve without a calculator: x = log₈4 + log₈16.
Solve without a calculator: logx + log(x - 3) = 1.
Solve without a calculator: logx + log(x - 3) = 1.
Solve with calculator: ln(x⁻⁴) = 7.
Solve with calculator: ln(x⁻⁴) = 7.
Solve with calculator: log(x - 3) = 1.
Solve with calculator: log(x - 3) = 1.
Solve word problem: Find the final amount when $10,000 is invested for 3 years at 3%. The interest is compounded continuously using the formula A = Pe^(rt).
Solve word problem: Find the final amount when $10,000 is invested for 3 years at 3%. The interest is compounded continuously using the formula A = Pe^(rt).
Flashcards
Transcendental Function
Transcendental Function
A function that cannot be expressed as a finite combination of algebraic operations.
One-to-One Function
One-to-One Function
A function where each input (x) has a unique output (y), and vice versa. It passes the horizontal line test.
Horizontal Translation
Horizontal Translation
The horizontal shift/slide of a function.
Logarithmic Power Property
Logarithmic Power Property
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Argument of a Logarithm
Argument of a Logarithm
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Line of Symmetry for Inverse Functions
Line of Symmetry for Inverse Functions
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Inverse of a Logarithmic Function
Inverse of a Logarithmic Function
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Expanding Logarithmic Expressions
Expanding Logarithmic Expressions
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Expanding the expression ln(xⁿ)
Expanding the expression ln(xⁿ)
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Condensing a Logarithmic Expression
Condensing a Logarithmic Expression
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Change of Base Formula
Change of Base Formula
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Finding Inverse Functions
Finding Inverse Functions
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Solving Exponential Equations
Solving Exponential Equations
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Logarithmic Sum Property
Logarithmic Sum Property
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Solving Logarithmic Equations
Solving Logarithmic Equations
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Continuous Compound Interest
Continuous Compound Interest
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Study Notes
Transcendental Functions
- Logarithmic functions qualify as transcendental functions, unlike polynomial, radical, or rational functions.
One-to-One Functions
- The function ƒ(x) = x² is not a one-to-one function, whereas ƒ(x) = x, ƒ(x) = x - 4, and ƒ(x) = x³ are one-to-one.
Function Values
- For ƒ(x) = 3^(x-1) and x = 4, the value is 27.
- For f(x) = 2e^(x) and x = 4, the rounded value is 109.20.
Transformations of Functions
- The number 2 in the equation f(x) = 3^(x-2) represents a horizontal translation.
Logarithmic Properties
- The expansion log x³ = 3 log x uses the power property of logarithms.
- The argument in the logarithmic function f(x) = log₂4 is 4.
Inverse Functions
- Inverse functions are symmetric to the line y = x.
- The inverse of a logarithmic function is an exponential function.
Logarithmic Expression Rewriting
- Expand log(xy/z) to log x + log y - log z.
- Expand lnx²y³ to 2ln x + 3ln y.
- Condense 3 log x - 4 log y to log(x³/y⁴).
- Condense ½ln x + ln y to ln(y√x).
Change of Base Formula
- The change of base formula can be used to evaluate logarithms, e.g., log₅11 or ln₄200 (showing work required).
Inverse Function Identification
- For f(x) = x³ + 9, derive the inverse function.
- For f(x) = (2x + 5)/x, derive the inverse function.
Exponential Equations
- To solve equations such as 8^(x² + 4) = 8¹³, equate the exponents.
- For log₈4 + log₈16, simplify using properties of logarithms.
Solving Logarithmic Equations
- Solve logx + log(x - 3) = 1 without a calculator by combining logarithms.
- For ln(x^-4) = 7, use a calculator for the solution, rounding to three decimal places.
- Solve x = log(x - 3) = 1 using a calculator and round to three decimal places.
Word Problems Involving Continuous Compounding
- To find the final amount when $10,000 is invested for 3 years at an interest rate of 3%, use the formula A = Pe^(rt) and round to two decimal places.
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