Chapter 8.5, Problem 61HP

(a)

To determine

To write: A logarithmic expression for a product and quotient condition and then write an expanded expression.

Answer to Problem 61HP

The expression for Product property is ${\log }_2\frac{10}{5}+{\log }_{2}2$ , expression for Quotient property is ${\log }_{2}10-{\log }_{2}5$ and expanded form is ${\log }_{2}10-{\log }_{2}5+{\log }_{2}2={\log }_2\left(\frac{10}{5}\cdot 2\right)$ .

Explanation of Solution

Given:

The given conditions are product and quotient conditions.

Calculation:

Expression for Quotient property can be written as follows.

  ${\log }_{2}10-{\log }_{2}5$

Expression for Product property can be written as follows.

  ${\log }_2\frac{10}{5}+{\log }_{2}2$

Expanded expression is as follows.

  ${\log }_{2}10-{\log }_{2}5+{\log }_{2}2={\log }_2\left(\frac{10}{5}\cdot 2\right)$

(b)

To determine

To write: A logarithmic expression for a product and power condition and then write an expanded expression.

Answer to Problem 61HP

The expression for Product property is ${\log }_{5}x+{\log }_{5}y$ , expression for Power property is $3{\log }_{5}xy$ and expanded form is $3\left({\log }_{5}x+{\log }_{5}y\right)={\log }_5{\left(xy\right)}^3$ .

Explanation of Solution

Given:

The given conditions are product and power conditions.

Calculation:

Expression for Product property can be written as follows.

  ${\log }_{5}x+{\log }_{5}y$

Expression for Power property can be written as follows.

  $3{\log }_{5}xy$

Expanded expression is as follows.

  $3\left({\log }_{5}x+{\log }_{5}y\right)={\log }_5{\left(xy\right)}^3$

(c)

To determine

To write: A logarithmic expression for a product, Quotient and power condition and then write an expanded expression.

Answer to Problem 61HP

The expression for Product property is ${\log }_{5}x+{\log }_{5}y$ , expression for Quotient property is ${\log }_{5}xy-{\log }_{5}z$ , expression for power property is $5{\log }_5\frac{xy}{z}$ and expanded form is $5\left({\log }_{5}x+{\log }_{5}y-{\log }_{5}z\right)={\log }_5{\left(\frac{xy}{z}\right)}^5$ .

Explanation of Solution

Given:

The given conditions are product, Quotient and power conditions.

Calculation:

Expression for Product property can be written as follows.

  ${\log }_{5}x+{\log }_{5}y$

Expression for Power property can be written as follows.

  $5{\log }_5\frac{xy}{z}$

Expression for Quotient property can be written as follows.

  ${\log }_{5}xy-{\log }_{5}z$

Expanded expression is as follows.

  $5\left({\log }_{5}x+{\log }_{5}y-{\log }_{5}z\right)={\log }_5{\left(\frac{xy}{z}\right)}^5$