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 thenumber e. Napier alluded to e as a "special number" associated with his development ofthe theory of logarithms. It is Leonard Euler (circa 1720), however, that defined and usedthe 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 asa reference to the "exponential."!! Before proceeding to the next Lesson, it would be good to recall (as seen in Exploration3.0.1) that the logarithm function is the inverse of the exponential function in such a waythat ifthenlogbase exponent = answer,baseThus, if 10²= 100, then log₁0 (100) = 2.(answer) = exponent.Lesson 3.2: The Natural Logarithm Function as the Inverse of exThe irrational number e is used extensively in the mathematics associated with finance andeconomics. As you have discovered, the number can be derived from exploring a specialcase of the amortization formulantA = P(1 + # ) "₁nwhere 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 compoundedper year. You are encouraged to research the connection between e and the mathematicsof finance.The next Exploration considers the inverse of the exponential function y=f(x) = e. For thefunction f, the symbol f¹ will be used to denote the inverse of f. Recall, from Unit 1 thatif we consider f(x) = y, then the inverse function is written as f-¹ (v)=x. In the case of theexponential 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 someproperties that should be true of its inverse. Write eª = 1 and eb = m. Consider theproperty ea eb = ea+b. Use this to show that f-¹ (lm) =f-¹ (1) + ƒ-¹ (m) is a propertyof 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, withthe understanding that this is an inverse function, it is normal to write y = ln (x) orf(x)= ln(x).

Answered: Image /qna-images/answer/f3334345-c258-4e84-8c30-1b6ab56d76cb.jpg

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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