The natural exponential function is given by exponential function that has the Euler number, <em>e</em>, as the base
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The correct options are as follows:
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 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
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
The reasons why the above options are correct are;
The natural exponential function is f(x) = , where <em>e</em> = Euler's number. It is the base of the natural logarithm, therefore;
= y
f(y) =
f⁻¹(y) = y =
Which gives;
- The natural exponential is <u>the </u><u>reciprocal </u><u>of the natural logarithm: </u><u>F</u>
- The natural exponential is <u>the </u><u>inverse </u><u>of the natural logarithm: </u><u>T</u>
- The natural exponential is <u>the negative of the natural logarithm: </u><u>F</u>
The input of the natural logarithm is <em>y</em>, where <em>y</em> = , therefore, <em>y</em> is always positive, given that <em>e</em><em> </em>is positive
Therefore;
- The domain of the natural logarithm is <u>the set of all positive numbers: </u><u>T</u>
- The domain of the natural logarithm is <u>the set of all </u><u>real </u><u>numbers: </u><u>F</u>
Given that <em>x</em> in , can be both positive, negative, fraction, irrational or zero, we have;
- The domain of the natural exponential is <u>the set of all </u><u>positive numbers</u><u>: </u><u>F</u>
- The domain of the natural exponential is <u>the set of all </u><u>real numbers</u><u>: </u><u>T</u>
Learn more about the exponential function here:
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