• The exponential function, exp : R → R, is the unique function f that satisfies f'(x) = f(x) for all x ER and f(0) = 1. o The natural logarithm, log, is the function inverse to the exponential function, exp: that is, for all x E R and for all y > 0, y = exp (x) if and only if x = log (y). a. Compute the definite integral xa-1 dx. b. Using your result from part a. and any appropriate rules for computing limits, show that lim / r exp t ra-l dx = t. a→0 You may use [exp (t)]ª exp (at) for all real a without proof. %3D c. Explain how, from your answers to parts a. and b., we may justify x- dx = log s.

Q: For any real number b > 1, and any real numbers x True False log, (x - y) = > 0 and y> 0, it must be…

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Q: Determine the nature of the following function for even and odd

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Q: X* Q: Prove that the function y- 2 2 x+1+/log Vx+Jx +1 + log satisfies the relationship 2y=x dx

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Q: Find the inverse of function g(x)=log, (log 3 x). y = 32* y = 23* y = 8* y = 9*

A: The given function is g(x)=log2(log3x)

Q: . Show that log(1) = 2nti, n = 0, ±1, ±2, ...

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Q: For the functions ƒ(x) = log2 (x) and g(x) = log6 (x), where is ƒ > g and g > ƒ?

A: The given functions fx=log2x and gx=log6x We have to find the intervals for which f>g and g>f.

Q: Let ƒ(x) = log₂(x + 1) and g(x) = log₂ (z − 1). Find h(x) = f(x) + g(x). - O a) h(z) = log₂ (21) Ob)…

A: This question is related to Functions, we will solve it using given information.

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A: Given the logarithm function f(x)=-2log(2x-8)

Q: 4. The domain of the logarithmic function f(x) = log, x is

A: fx=logax

Q: Show that log,(n) € O(log,(n)) for any b>0 with b 1· Generalize this result.

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Q: The inverse function of f(x) = 2 log3 (1-x) is اختراحد الخيارات A. 3 -1o B. 1+ 3 C. 3 D. 1-3

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Q: Pls helpp ASAP

A: Given  f(x) = log2 (x+1)   and g(x) = log2 (x-1)

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Q: Let g (x) = log,, (x). Use this function to answer each ques a) g (16) = b) g (2) = c) g (0) = %3D

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Q: The function f(x) interval [1, e] at x = Hora lo 0 x is the nat

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Q: Determine the nature of the following function for even and odd f(x)=log(x+vx°

A: if f(-x) = f(x) then f is even  if f(-x) = -f(x) then f is odd if if f(-x) is not equal to f(x) and…

Q: According to the properties of logarithms In sec x= - In cos x O True O False

A: Since you have asked multiple question, we will solve the first question for you. If you want any…