Question

(a) How is the logarithmic function \(y = {\log _b}x\) defined?

(b) What is the domain of this function?

(c) What is the range of this function?

(d) Sketch the general shape of the graph of the function \(y = {\log _b}x\) if \(b > {\bf{1}}\).

Short answer

a) The logarithmic function \(y = {\log _b}x\) is defined as the inverse of the exponential function with base \(b\) and is denoted by \({\log _b}x = y \Leftrightarrow {b^y} = x\).

b) The domain of the function is \(\left( {0,\infty } \right)\).

c) The range of the function is the set of all real numbers or \(\mathbb{R}\).

d) The graph of the function \(y = {\log _b}x\) is shown below:

Step-by-step solution

Logarithmic function

The exponential function \(f\left( x \right) = {b^x}\) isone-to-one by the Horizontal Line Test. Therefore, it has an inverse function \({f^{ - 1}}\) known as thelogarithmic function with base\(b\) , and it is denoted by \({\log _b}\). If the formulation of an inverse function is given by \({f^{ - 1}}\left( x \right) = y \Leftrightarrow f\left( y \right) = x\) then \({\log _b}x = y \Leftrightarrow {b^y} = x\).

Explain the logarithmic function \(y = {\log _b}x\) defined

a)

The logarithmic function \(y = {\log _b}x\) is defined as the inverse of the exponential function with base \(b\) and is denoted by \({\log _b}x = y \Leftrightarrow {b^y} = x\).

Determine the domain of the function

b)

The domain of the function \(y = {\log _b}x\) is \(\left( {0,\infty } \right)\).

Determine the range of the function

c)

The range of the function is the set of all real numbers or \(\mathbb{R}\).

Sketch the general shape of the graph of the function \(y = {\log _b}x\)

d)

Sketch the graph of the function \(y = {\log _b}x\) as shown below:

Thus, the graph is obtained.