#### To determine

**To find:**

(a) How is the logarithmic function y=logbx

(b) Domain of the function y=logbx.

(c) Range of the function y=logbx.

(d) General shape of the graph of the function y=logbx if b>1

#### Answer

(a) The logarithmic function y=logbx is defined by by=x.

(b) The logarithmic function y=logbx has domain (0,∞).

(c) The logarithmic function y=logbx has range *R* (set of real numbers)

(d) See the graph

(a)

**Given data:**

Logarithmic function y=logbx.

#### Explanation

If b>0 and b≠1, the exponential function f(x)=bx is either increasing or decreasing and so it is one to one. It therefore has an inverse function f−1, which is called logarithmic function with base b and is denoted by logb. If we use the formulation of an inverse function

f−1(x)=y ⇔ f(y)=x

**Calculation:**

Therefore, logbx=y⇔x=by

**(b)**

**Given data:**

Logarithmic function y=logbx.

The logarithmic function y=logbx has domain (0, ∞) since in this function x can take any value from the interval (0, ∞)

**(c)**

**Given data:**

Logarithmic function y=logbx.

The logarithmic function y=logbx has range R since in this function *x* can take any value from the interval (0, ∞) and values of y are from set R

**(d)**

**Given data:**

Logarithmic function y=logbx.

The logarithmic function y=logbx is an increasing function, it has domain (0, ∞) since in this function *x* can take any value from the interval (0, ∞) and values of *y* are from set R.

**Conclusion:**

(a) The logarithmic function y=logbx is defined by by=x.

(b) The logarithmic function y=logbx has domain (0,∞).

(c) The logarithmic function y=logbx has range *R* (set of real numbers)

(d) See the graph

(e)