#### To determine

**(a)**

How is the number *e* defined.

#### Answer

e=limx→∞(1+1n)n.

#### Explanation

**Given:** A number *e*.

The number *e* can be defined a few ways. *e* is defined in this book as the number such that

limx→∞en−1n=1.

Later on, one can observe:

e=∑n=0∞1n!=limx→∞(1+1n)n

**Conclusion:** The number *e* is defined as e=limx→∞(1+1n)n.

#### To determine

**(b)**

Express *e* as a limit.

#### Answer

e=limx→∞(1+1n)n.

#### Explanation

**Given:** The number is *e*.

*e* is defined as a limit.

limx→∞en−1n=1

e=∑n=001n!=limx→∞(1+1n)n

**Conclusion:** Hence, *e* as a limit is e=limx→∞(1+1n)n.

#### To determine

**(c)**

Why is the natural logarithm function y=ex used, more often in calculus than the other exponential function y=bx.

#### Answer

The natural exponential function is its own derivative.

#### Explanation

**Given:** y=ex,y=bx.

When one use axx however, the derivative would be axlna, which adds another function lna

This make take its integral slightly harder as well.

So y=ex is more used than y=bx.

**Conclusion:** Hence, the natural exponential function is its own derivative other exponential function are not. So y=ex is more used than y=bx.

#### To determine

**(d)**

Why is the natural logarithm function y=lnx used more often in the other logarithmic function y=logx.

#### Answer

The natural exponential function is its own derivative. Other exponential function is not.

#### Explanation

**Given:** y=lnx and y=logbx.

Well the good thing about ex is that the ‘*ln*’ function is based off of it. So lne=1,lna=??

One would need a calculator for doing that, when one have a calculator problem, such as taking the derivative of ex, it would be itself, ex=lne, which is simply ex because lne=1

Which mean its integral is also ex.

**Conclusion:** Hence, the natural exponential function is its own derivative, other exponential function is not so y=lnx is more used other than u=logbx.