#### To determine

the domain and range of the natural exponential function f(x)=ex

#### Answer

**Solution**: Domain is (−∞,∞).

Range is (0,∞).

#### Explanation

**Given**: f(x)=ex

You may draw the graph of f(x)=ex

The key things to notice is that f(x) has domain D:(−∞,∞) and range R:(0,∞) this implies that f(x) has a horizontal asymptote y=0.

ex will get intentionally close to the line y=0, but it will never actually reach that time. Thus it has no x intercepts.

**Conclusion:** Hence, the domain of ex is D:(−∞,∞) and range of exR:(0,∞).

#### To determine

the domain and range of the natural logrithmic function g(x)=lnx

#### Answer

**Solution**: domain is (0,∞).

Range is (−∞,∞).

#### Explanation

**Given**:g(x)=lnx.

Draw the graph of g(x)=lnx look like

The domain of this function is

Domain: (0,∞)

and Range: (−∞,∞)

**Conclusion:** Hence, the domain and range of g(x)=lnx

Domain:(0,∞)

Range :(−∞,∞).

#### To determine

How are the function f(x)=ex and g(x)=lnx related.

#### Answer

**Solution**: Symmetrical with respect to the line y=x.

#### Explanation

Let f(x)=ex and g(x)=lnx and show that f and g satisfy the inverse relations.

**Given**: f(x)=ex,g(x)=lnx.

Draw the graph of y=ex and y=lnx

As with all pairs of inverse functions, their graphs are symmetrical with respect to the line y=x.

**Conclusion:** Hence, f(x)=ex and g(x)=lnx are the symmetrical with respect to the line y=x.

#### To determine

How are the graph of these function related? Sketch these graph by hand, using the same axies.

#### Answer

**Solution**: Symmetrical with respect to the line y=x.

#### Explanation

Let f(x)=ex and g(x)=lnx and show that f and g satisfy the inverse relations.

**Given**: f(x)=ex,g(x)=lnx.

Draw the graphs of y=ex and y=lnx

As with all pairs of inverse functions, their graphs are symmetrical with respect to the line y=x.

**Conclusion:** Hence, the relation f(x)=ex and g(x)=lnx is symmetrical with respect to the line y=x.

#### To determine

If b is a positive number b≠1, write an equation that expresses logbx in terms of lnx.

#### Answer

**Solution**: lnx=logbx or lnx=logb∗logx.

#### Explanation

If b is a positive number and b≠1 then the logbx is expresses in terms of lnx is

**Given**: b>0 and b≠1.

lnx=logbxlnx=logx∗logblnx=logb∗logx

**Conclusion:** Hence, the logbx in terms of lnx is lnx=logbx or lnx=logb∗logx.