Chapter 8.5, Problem 63HP

(a)

To determine

To explain: The reason why ${\log }_{b}1=0$ .

Answer to Problem 63HP

The equation ${\log }_{b}1=0$ because $b^0=1$ .

Explanation of Solution

Given:

The given equation is ${\log }_{b}1=0$ .

Calculation:

Consider the expression.

  ${\log }_{b}1=0$

The equation ${\log }_{b}1=0$ because $b^0=1$ .

(b)

To determine

To explain: The reason why ${\log }_{b}b=1$ .

Answer to Problem 63HP

The equation ${\log }_{b}b=1$ because $b^1=b$ .

Explanation of Solution

Given:

The given equation is ${\log }_{b}b=1$ .

Calculation:

Consider the expression.

  ${\log }_{b}b=1$

The equation ${\log }_{b}b=1$ because $b^1=b$ .

(c)

To determine

To explain: The reason why ${\log }_b{b}^x=x$ .

Answer to Problem 63HP

The equation ${\log }_b{b}^x=x$ because $b^x=b^x$ .

Explanation of Solution

Given:

The given equation is ${\log }_b{b}^x=x$ .

Calculation:

Consider the expression.

  ${\log }_b{b}^x=x$

From the power rule,

  $\begin{array}{c}{\log }_b{b}^x=x\\ x{\log }_{b}b=x\\ x=x\end{array}$

The equation ${\log }_b{b}^x=x$ because $b^x=b^x$ .