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For the following statements, enter T if they are true and F if they are false. Logarithms and
exponentials can be defined for complex numbers, and then can assume complex function values, but
in the questions below concerning domain and range, let's consider only real numbers.
The natural exponential is the reciprocal of the natural logarithm.
The natural exponential is the inverse of the natural logarithm.
The natural exponential is the negative of the natural logarithm.
The domain of the natural logarithm is the set of all positive numbers.
The domain of the natural logarithm is the set of all real numbers.
The domain of the natural exponential is the set of all positive numbers.
The domain of the natural exponential is the set of all real numbers.

Respuesta :

The natural exponential function is given by exponential function that has the Euler number, e, as the base

The correct options are as follows:

  • The natural exponential is the reciprocal of the natural logarithm: F (False)
  • The natural exponential is the inverse of the natural logarithm: T (True)
  • The natural exponential is the negative of the natural logarithm: F (False)
  • The domain of the natural logarithm is the set of all positive numbers: T (True)
  • The domain of the natural logarithm is the set of all real numbers: F (False)
  • The domain of the natural exponential is the set of all positive numbers: F (False)
  • The domain of the natural exponential is the set of all real numbers: T (True)

The reasons why the above options are correct are;

The natural exponential function is f(x) = [tex]e^x[/tex], where e = Euler's number. It is the base of the natural logarithm, therefore;

[tex]log_e(y) = x[/tex]

[tex]e^x[/tex] = y

[tex]e^x \neq log_e(y) = x[/tex]

f(y) = [tex]log_e(y)[/tex]

f⁻¹(y) = y = [tex]e^x[/tex]

Which gives;

  • The natural exponential is the reciprocal of the natural logarithm: F
  • The natural exponential is the inverse of the natural logarithm: T
  • The natural exponential is the negative of the natural logarithm: F

The input of the natural logarithm is y, where y = [tex]e^x[/tex], therefore, y is always positive, given that e is positive

Therefore;

  • The domain of the natural logarithm is the set of all positive numbers: T
  • The domain of the natural logarithm is the set of all real numbers: F

Given that x in  [tex]\mathbf{e^x}[/tex], can be both positive, negative, fraction, irrational or zero, we have;

  • The domain of the natural exponential is the set of all positive numbers: F
  • The domain of the natural exponential is the set of all real numbers: T

Learn more about the exponential function here:

https://brainly.com/question/21730432

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