Chapter 6.5, Problem 47E

To determine

Tocalculate:The change of base formula using the properties of exponents.

Answer to Problem 47E

The required formula is expressed in change of base form as $z=\frac{x}{y}$ .

Explanation of Solution

Given information:

The given values of x, y and z are:

  $x={\log }_{b}a,y={\log }_{b}c,$

  $z={\log }_{c}a$

Formula used:

Product property of logarithm:

  ${\log }_{b}mn={\log }_{b}m+{\log }_{b}n$

Consider thegiven value of x.

  $x={\log }_{b}a$

Express the above expression in exponent form using the properties of exponent of logarithmic functions.

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

Consider the given value of y.

  $y={\log }_{b}c$

Express the above expression in exponent form using the properties of exponent of logarithmic functions.

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

Consider the given value of z.

  $z={\log }_{c}a$

Express the above expression in exponent form using the properties of exponent of logarithmic functions.

  $\begin{array}{c}z={\log }_{c}a\\ a=c^z\end{array}$

From the above values of a, b and c, it is seen that:

  $\begin{array}{c}{\left(b^y\right)}^z=b^x=a\\ yz=x\\ z=\frac{x}{y}\end{array}$

Hence, the formula is expressed in change of base format by $z=\frac{x}{y}$ .