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Explain how you know that the exponential function has an Knowing the properties of the exponential function can help us decide on some properties that should be true of its inverse. Write eª = 1 and eb = m. Consider the property ea eb = ea+b. Use this to show that f-¹ (lm) =f¹ (1) +f-¹ (m) is a property of the inverse function.

Explain how you know that the exponential function has an Knowing the properties of the exponential function can help us decide on some properties that should be true of its inverse. Write eª = 1 and eb = m. Consider the property ea eb = ea+b. Use this to show that f-¹ (lm) =f¹ (1) +f-¹ (m) is a property of the inverse function.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I need help on the 2 parts I don’t really understand it(background info in one pic and the parts of it are in the other pic)
Historical Note: The mathematician John Napier (circa 1600) is credited as being the first to introduce the number e. Napier alluded to e as a "special number" associated with his development of the theory of logarithms. It is Leonard Euler (circa 1720), however, that defined and used the symbol e to refer to Napier's "special number." Euler also discovered many of this num- ber's special properties. It is likely that Euler chose the symbol e for this natural number as a reference to the "exponential." !! Before proceeding to the next Lesson, it would be good to recall (as seen in Exploration 3.0.1) that the logarithm function is the inverse of the exponential function in such a way that if then log base exponent = answer, base Thus, if 10²= 100, then log₁0 (100) = 2. (answer) = exponent. Lesson 3.2: The Natural Logarithm Function as the Inverse of ex The irrational number e is used extensively in the mathematics associated with finance and economics. As you have discovered, the number can be derived from exploring a special case of the amortization formula nt A = P(1 + # ) "₁ n where A is the total value of an investment after t years, P is the initial or principal invest- ment, r is the rate of interest, and n is the number of times that the interest is compounded per year. You are encouraged to research the connection between e and the mathematics of finance. The next Exploration considers the inverse of the exponential function y=f(x) = e. For the function f, the symbol f¹ will be used to denote the inverse of f. Recall, from Unit 1 that if we consider f(x) = y, then the inverse function is written as f-¹ (v)=x. In the case of the exponential function, we can write f (y)=f(e)=x.
Transcribed Image Text:Historical Note: The mathematician John Napier (circa 1600) is credited as being the first to introduce the number e. Napier alluded to e as a "special number" associated with his development of the theory of logarithms. It is Leonard Euler (circa 1720), however, that defined and used the symbol e to refer to Napier's "special number." Euler also discovered many of this num- ber's special properties. It is likely that Euler chose the symbol e for this natural number as a reference to the "exponential." !! Before proceeding to the next Lesson, it would be good to recall (as seen in Exploration 3.0.1) that the logarithm function is the inverse of the exponential function in such a way that if then log base exponent = answer, base Thus, if 10²= 100, then log₁0 (100) = 2. (answer) = exponent. Lesson 3.2: The Natural Logarithm Function as the Inverse of ex The irrational number e is used extensively in the mathematics associated with finance and economics. As you have discovered, the number can be derived from exploring a special case of the amortization formula nt A = P(1 + # ) "₁ n where A is the total value of an investment after t years, P is the initial or principal invest- ment, r is the rate of interest, and n is the number of times that the interest is compounded per year. You are encouraged to research the connection between e and the mathematics of finance. The next Exploration considers the inverse of the exponential function y=f(x) = e. For the function f, the symbol f¹ will be used to denote the inverse of f. Recall, from Unit 1 that if we consider f(x) = y, then the inverse function is written as f-¹ (v)=x. In the case of the exponential function, we can write f (y)=f(e)=x.
Explain how you know that the exponential function has an inverse. Knowing the properties of the exponential function can help us decide on some properties that should be true of its inverse. Write eª = 1 and eb = m. Consider the property ea eb = ea+b. Use this to show that f-¹ (lm) =f-¹ (1) + ƒ-¹ (m) is a property of the inverse function. The inverse function we've been exploring is called the natural logarithm, and is writ- ten as In (y). Rather than writing f-¹(y) = x, we can write In (y) = x; however, with the understanding that this is an inverse function, it is normal to write y = ln (x) or f(x)= ln(x).
Transcribed Image Text:Explain how you know that the exponential function has an inverse. Knowing the properties of the exponential function can help us decide on some properties that should be true of its inverse. Write eª = 1 and eb = m. Consider the property ea eb = ea+b. Use this to show that f-¹ (lm) =f-¹ (1) + ƒ-¹ (m) is a property of the inverse function. The inverse function we've been exploring is called the natural logarithm, and is writ- ten as In (y). Rather than writing f-¹(y) = x, we can write In (y) = x; however, with the understanding that this is an inverse function, it is normal to write y = ln (x) or f(x)= ln(x).
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