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2. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please email: educators@boundless.com http://boundless.com/teaching-platform

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4. Logarithmic Functions Special Logarithms Converting between Exponential and Logarithmic Equations Natural Logarithms Changing Logarithmic Bases Graphs of Logarithmic Functions Solving Problems with Logarithmic Graphs Graphing Logarithmic Functions Exponents and Logarithms > Graphing Logarithmic Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The inverse of the logarithmic operation is exponentiation. The logarithm is commonly used in many fields: that with base 2 in computer science, that with base e in pure mathematics, and that with base 10 in natural science and engineering. The logarithm of a product is the sum of the logarithms of the factors. Logarithmic Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/logarithmic- functions-174- 5869?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

6. Logarithms with base equal to 10 are called common logarithms. They are most applicable in physical and natural sciences and engineering. Logarithms with base equal to e are called natural logarithms. They are most applicable in pure mathematics. Logarithms with base equal to 2 are called binary logarithms. They are most applicable in computer science. Special Logarithms Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/special- logarithms-175- 11093?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

7. The logarithmic and exponential operations are inverses. If given an exponential equation, one can take the natural logarithm to isolate the variables of interest, and vice versa. Converting from logarithmic to exponential form can make for easier equation solving. Converting between Exponential and Logarithmic Equations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/converting- between-exponential-and-logarithmic-equations-176- 11086?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

8. The natural logarithm is the logarithm with base equal to e. Also known as Euler's number, e is an irrational number that often appears in natural relationships in pure math and science. The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. Natural Logarithms Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/natural- logarithms-177- 5881?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

9. If a logarithm and/or its base are not whole numbers, evaluation can be near- impossible. One can change the base of a logarithm by expressing it as the quotient of two logarithms with a common, same base. Changing a logarithm's base to 10 makes it much simpler to evaluate; it can be done on a calculator. Changing Logarithmic Bases Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/changing- logarithmic-bases-178- 5890?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

10. The logarithmic graph is similar in shape to the square root graph, but with a vertical asymptote as x approaches 0 from the right. With use of defined numbers, logarithmic graphs can be shifted horizontally and/or vertically. When graphing logarithms, it is usually advisable to use several x values between the vertical asymptote and the next integer. After the first integer, it is usually best to make points less and less frequently. Graphs of Logarithmic Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/graphs-of- logarithmic-functions-179- 5891?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

11. Logarithmic graphs use logarithmic scales, in which the values differ exponentially. For example, instead of including marks at 0, 1, 2, and 3, a logarithmic scale may include marks at 0.1, 1, 10, and 100, each an equal distance from the previous and next. Logarithmic graphs allow one to plot a very large range of data without losing the shape of the graph. Logarithmic graphs make it easier to interpolate in areas that may be difficult to read on linear axes. For example, if the plot of y=x5 is scaled to show a very wide range of y values, the curvature near the origin may be indistinguishable on linear axes. It is much clearer on logarithmic axes. Solving Problems with Logarithmic Graphs Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/graphing-logarithmic-functions-37/solving- problems-with-logarithmic-graphs-180- 5878?campaign_content=book_196_section_37&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Logarithmic Scale View on Boundless.com Exponents and Logarithms > Graphing Logarithmic Functions

12. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

13. Key terms asymptote A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity". base A number raised to the power of an exponent. dependent variable The variable in an equation or function whose value depends on one or more variables in the equation or function. e The base of the natural logarithm, 2.718281828459045… exponent The power to which a number, symbol, or expression is to be raised. For example, the 3 in x^3. exponentiation The process of calculating a power by multiplying together a number of equal factors, where the exponent specifies the number of factors to multiply. independent variable Any variable in an equation or function whose value is not dependent on any other in the equation or function. interpolate To estimate the value of a function between two points between which it is tabulated. logarithm The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

14. logarithm The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. logarithmic function any function in which an independent variable appears in the form of a logarithm; they are the inverse functions of exponential functions natural logarithm The logarithm in base e; either the function that given x\! returns y\! such that e^y = x\!, or the value of y\!. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

15. Interactive Graph: Graph of the Natural Logarithm Graph of a natural logarithm with the equation. The natural logarithm is the logarithm with base equal to Euler's number, e. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of the Natural Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm View on Boundless.com Exponents and Logarithms

16. Interactive Graph: The Exponential Function Graph of the natural exponential function. Notice how the graph changes depending on the way in which you play with the variants. How would you inverse the graph to make it look like the other graph in this chapter? Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: The Exponential Function." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-the-exponential-function View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-the-exponential-functionView on Boundless.com Exponents and Logarithms

17. Interactive Graph: Graph of the Natural Logarithm Graph of a natural logarithm. The equation for this graph can be expressed as either or. Try using either equation and notice how they are identical. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of the Natural Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm- d3143c04-59c6-4833-bc3b-cb093a4d3586 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm- d3143c04-59c6-4833-bc3b-cb093a4d3586View on Boundless.com Exponents and Logarithms

18. Interactive Graph: Graph of the Natural Logarithm Graph of the natural logarithm, or. The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0. What shape does the graph become? Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of the Natural Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm- e8371179-8674-42a1-b624-6ce3f4c8e9be View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-the-natural-logarithm- e8371179-8674-42a1-b624-6ce3f4c8e9beView on Boundless.com Exponents and Logarithms

19. Interactive Graph: Graph of Binary Logarithm Graph of a binary logarithm,. This function can be expressed as the quotient of log (x) and log (2), both with a base of 10. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Binary Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithm- 74642101-fe32-425d-93c0-74b01c8b8248 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithm- 74642101-fe32-425d-93c0-74b01c8b8248View on Boundless.com Exponents and Logarithms

20. Interactive Graph: Graph of Binary Logarithm Graph of binary logarithm with the equation. The function's slope decreases with increasing x, thus containing its vertical growth. More importantly, the scale is greatly reduced, allowing for easy visualization of large values of y. In many cases, by taking the logarithms of exponential equations, the equations become linear, and plotting the appropriate logarithms can help solve for roots, slopes, and intercepts. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Binary Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithm View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithmView on Boundless.com Exponents and Logarithms

21. Logarithmic Scale The graphs of functions f(x)=10^x, f(x)=x, and f(x)=log(x) on four different coordinate plots. Note how each function changes shape on each set of coordinates. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Logarithmic Scales." CC BY-SA http://en.wikipedia.org/wiki/File:Logarithmic_Scales.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Logarithmic_Scales.svgView on Boundless.com Exponents and Logarithms

22. Graph of log(y) on a Semi-Log Scale Both plots capture y well for their respective ranges, but note how easily distinguibile the points are in both the lower and higher areas. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "SemiLogPlotDescription." CC BY-SA http://en.wikipedia.org/wiki/File:SemiLogPlotDescription.GIF View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:SemiLogPlotDescription.GIFView on Boundless.com Exponents and Logarithms

23. Interactive Graph: Logarithms Graph of logarithm functions. All logarithmic functions will have the same general shape, but their graphs will vary depending on base and any coefficients inserted into the equation. In this figure, there is the natural log ln(x) which is log base e (red) and log(x) base 10 (blue). Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Logarithms." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-logarithms View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-logarithmsView on Boundless.com Exponents and Logarithms

24. Points of log(y) on a Linear Scale Notice how values of y less than 10 are indistinguibile. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "LargeSpanLinearPlot." CC BY-SA http://en.wikipedia.org/wiki/File:LargeSpanLinearPlot.GIF View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:LargeSpanLinearPlot.GIFView on Boundless.com Exponents and Logarithms

25. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is NOT true of logarithmic functions? A) A logarithm with no base written is assumed to have base 2. B) A logarithm is an exponent. C) A logarithm with base e is called the natural logarithm. D) The inverse of the logarithmic operation is exponentiation.

26. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is NOT true of logarithmic functions? A) A logarithm with no base written is assumed to have base 2. B) A logarithm is an exponent. C) A logarithm with base e is called the natural logarithm. D) The inverse of the logarithmic operation is exponentiation.

27. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Common logarithms are often used in: A) computer science with the use of integral parts. B) the natural sciences and engineering. C) the physical sciences and pure math. D) none of these answers.

28. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Common logarithms are often used in: A) computer science with the use of integral parts. B) the natural sciences and engineering. C) the physical sciences and pure math. D) none of these answers.

29. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms A logarithm with base e is called: A) a common logarithm. B) a binary logarithim. C) none of these answers. D) a natural logarithm.

30. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms A logarithm with base e is called: A) a common logarithm. B) a binary logarithim. C) none of these answers. D) a natural logarithm.

31. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Binary logarithms are commonly used in: A) the natural sciences and engineering. B) computer science with the use of integral parts. C) the physical sciences and pure math. D) none of these answers.

32. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Binary logarithms are commonly used in: A) the natural sciences and engineering. B) computer science with the use of integral parts. C) the physical sciences and pure math. D) none of these answers.

33. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms What is 34=81 in logarithmic form? A) log381=4 B) log481=3 C) log813=4 D) log34=81

34. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms What is 34=81 in logarithmic form? A) log381=4 B) log481=3 C) log813=4 D) log34=81

35. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms What is 82=64 in logarithmic form? A) log264=8 B) log28=64 C) log642=8 D) log864=2

36. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms What is 82=64 in logarithmic form? A) log264=8 B) log28=64 C) log642=8 D) log864=2

37. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Euler's number, also known as e, can be represented by: A) the limit of (1+1/n)n B) iπ C) log10(2) D) none of these answers

38. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Euler's number, also known as e, can be represented by: A) the limit of (1+1/n)n B) iπ C) log10(2) D) none of these answers

39. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is equivalent to log6(7)? A) [log10(7)]/[log10(6)] B) [log10(6)]/[log10(7)] C) [log2(7)]/[log2(6)] D) [log2(6)]/[log2(7)]

40. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is equivalent to log6(7)? A) [log10(7)]/[log10(6)] B) [log10(6)]/[log10(7)] C) [log2(7)]/[log2(6)] D) [log2(6)]/[log2(7)]

41. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms What is the decimal approximation of log3(5)? A) 0.683 B) 1.465 C) 1.099 D) 1.609

42. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms What is the decimal approximation of log3(5)? A) 0.683 B) 1.465 C) 1.099 D) 1.609

43. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms If we wanted to shift the graph of y=log(x) down by 3 units, what would be the resulting equation? A) y=log(x)+3 B) y=log(x-3) C) y=log(x)-3 D) y=log(x+3)

44. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms If we wanted to shift the graph of y=log(x) down by 3 units, what would be the resulting equation? A) y=log(x)+3 B) y=log(x-3) C) y=log(x)-3 D) y=log(x+3)

45. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following characteristics is TRUE of the graphs of logarithmic functions? A) The graph never crosses the x-axis. B) The slope of the graph gradually decreases from left to right. C) The graph crosses the y-axis. D) The domain and range are equal.

46. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following characteristics is TRUE of the graphs of logarithmic functions? A) The graph never crosses the x-axis. B) The slope of the graph gradually decreases from left to right. C) The graph crosses the y-axis. D) The domain and range are equal.

47. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms If we wanted to shift the graph of y=log(x) to the left by 2 units, what would be the resulting equation? A) y=log(x-2) B) y=log(x)+2 C) y=log(x)-2 D) y=log(x+2)

48. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms If we wanted to shift the graph of y=log(x) to the left by 2 units, what would be the resulting equation? A) y=log(x-2) B) y=log(x)+2 C) y=log(x)-2 D) y=log(x+2)

49. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is NOT an advantage of using logarithmic scales? A) They allow one to plot a very large range of data without losing shape of the graph. B) They allow one to interpolate at any point on the plot, regardless of the range of the graph. C) They help to make large ranges of data make more sense. D) They make the differences between points remain constant.

50. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is NOT an advantage of using logarithmic scales? A) They allow one to plot a very large range of data without losing shape of the graph. B) They allow one to interpolate at any point on the plot, regardless of the range of the graph. C) They help to make large ranges of data make more sense. D) They make the differences between points remain constant.

51. Attribution Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wiktionary. "exponentiation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentiationCC BY-SA 3.0http://en.wiktionary.org/wiki/exponentiation Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Wikipedia. "Logarithmic scale." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Logarithmic_scaleCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithmic_scale Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Wiktionary. "interpolate." CC BY-SA 3.0 http://en.wiktionary.org/wiki/interpolateCC BY-SA 3.0http://en.wiktionary.org/wiki/interpolate Wikipedia. "Natural logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Natural_logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Natural_logarithm Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wiktionary. "natural logarithm." CC BY-SA 3.0 http://en.wiktionary.org/wiki/natural+logarithmCC BY-SA 3.0http://en.wiktionary.org/wiki/natural+logarithm Wiktionary. "e." CC BY-SA 3.0 http://en.wiktionary.org/wiki/eCC BY-SA 3.0http://en.wiktionary.org/wiki/e Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//chemistry/definition/baseCC BY-SA 3.0http://www.boundless.com//chemistry/definition/base Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wiktionary. "logarithmic function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/logarithmic+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/logarithmic+function Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

52. Wiktionary. "asymptote." CC BY-SA 3.0 http://en.wiktionary.org/wiki/asymptoteCC BY-SA 3.0http://en.wiktionary.org/wiki/asymptote Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Wiktionary. "independent variable." CC BY-SA 3.0 http://en.wiktionary.org/wiki/independent+variableCC BY-SA 3.0http://en.wiktionary.org/wiki/independent+variable Wiktionary. "dependent variable." CC BY-SA 3.0 http://en.wiktionary.org/wiki/dependent+variableCC BY-SA 3.0http://en.wiktionary.org/wiki/dependent+variable Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wikipedia. "Binary logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Binary_logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Binary_logarithm Wikipedia. "Natural logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Natural_logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Natural_logarithm Wiktionary. "natural logarithm." CC BY-SA 3.0 http://en.wiktionary.org/wiki/natural+logarithmCC BY-SA 3.0http://en.wiktionary.org/wiki/natural+logarithm Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Wiktionary. "e." CC BY-SA 3.0 http://en.wiktionary.org/wiki/eCC BY-SA 3.0http://en.wiktionary.org/wiki/e Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms