Chapter EP, Problem 8.5.2EP

To determine

To find: The approximate value of expression of ${\log }_{3}245$ .

Answer to Problem 8.5.2EP

The approximate value of expression is ${\log }_{3}245=5.0075$

Explanation of Solution

Given information:

The given equations ${\log }_{3}5\approx 1.4651$ and ${\log }_{3}7\approx 1.7712$ .

Calculation:

Consider the expression.

${\log }_{3}245$ .

The logarithm of a quotient is the difference of the logarithms of the numerator and denominator, that is, for all positive numbers m, n and b , where $b\ne 1$ ,

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

The logarithm of a power is the product of the logarithm and the exponent.

For any real number p and positive numbers m and b, where $b\ne 1$ ,

${\log }_n{m}^p=p{\log }_{b}m$

Use ${\log }_{3}5\approx 1.4651$ and ${\log }_{3}7\approx 1.7712$ to approximate ${\log }_{3}245$ .

Calculate the logarithm.

$\begin{array}{c}{\log }_{3}245={\log }_3(5.49)\\ ={\log }_{3}5+{\log }_{3}49\\ ={\log }_{3}5+{\log }_3{7}^2\\ ={\log }_{3}5+2{\log }_{3}7\end{array}$

  $\begin{array}{c}{\log }_{3}245=\text{1}.\text{4651}+\text{2}\left(\text{1}.\text{7712}\right)\\ =\text{5}.00\text{75}\end{array}$

Therefore, the approximate value of expression is ${\log }_{3}245=5.0075$ .