Chapter 1.6, Problem 33E

(a)

To determine

To Explain: The definition of  the logarithmic function $y={\log }_{b}x$.

Explanation of Solution

Let $f\left(x\right)={\text{b}}^x$ be the exponential function and if $b>1$, the function is increasing, if $0<b<1$, the function is decreasing.

To prove:

$f\left(x\right)=f\left(y\right)\text{ iff }x=y$

Proof:

Consider $f\left(x\right)={\text{b}}^x$ and $f\left(y\right)={\text{b}}^y$

$\begin{array}{c}f\left(x\right)=f\left(y\right)\\ {\text{b}}^x={\text{b}}^y\end{array}$

Equating the powers on both sides,

$x=y$

Therefore, it is one to one and $\text{b}>1$ so $f^{-1}$ exists.

Here $f^{-1}$ is called logarithmic function with base b with respect to x.

Thus, the logarithm of x in base b is written for $x>0$ is given by $y={\log }_{b}x$ if an only if ${\text{b}}^y=x$ where $x>0$ and $b>0$.

(b)

To determine

To find: The domain of $y={\log }_{b}x$.

Answer to Problem 33E

The domain of $y={\log }_{b}x$ is the set of positive real numbers.

Explanation of Solution

Since the logarithmic function is the inverse of the exponential function, the domain of logarithmic function is the range of exponential function.

The exponential function is always positive so logarithmic function is not definedfor negative numbers or for zero.

Therefore, the domain of the logarithmic function $y={\log }_{b}x$ is the set of positive real numbers.

(c)

To determine

To find: The range of $y={\log }_{b}x$.

Answer to Problem 33E

Solution:

The range of $y={\log }_{b}x$ is set of all real numbers.

Explanation of Solution

Since the logarithmic function is the inverse of the exponential function, the range of logarithmic function is the domain of exponential function.

The exponential function is defined for all real numbers so logarithmic function can take any real number.

Therefore, the range of the logarithmic function $y={\log }_{b}x$ is the set of all real numbers.

(d)

To determine

To sketch: The graph of the function $y={\log }_{b}x$.

Explanation of Solution

Graph:

Use online graph calculator and draw the graph of the function $y={\log }_{b}x$ if $\text{b}>1$ as shown below in Figure 1.