Chapter 3.3, Problem 119E

To determine

How many numbers between 1 and 20 can be approximated using the given values of logarithms?

Answer to Problem 119E

Apart from 2, 3 and 5, nine more numbers can be approximated between 1 and 20 using the given values of logarithms.

Explanation of Solution

Given:

Three logarithms: $\ln 2=0.6931,\ln 3=1.0986,\ln 5=1.6094$ .

Calculation:

The following numbers can be approximated using the given three logarithms:

  $2,3,4,5,6,8,9,10,12,15,16,18$

Below are the calculations for each one of these numbers:

  $\begin{array}{l}\ln 2=0.6931\left(\text{Given}\right)\\ \ln 3=1.0986\left(\text{Given}\right)\\ \ln 4=\ln 2^2=2\ln 2=2\times 0.6931=1.3862\\ \ln 5=1.6094\left(\text{Given}\right)\\ \ln 6=\ln \left(2\times 3\right)=\ln 2+\ln 3=0.6931+1.0986=1.7917\\ \ln 8=\ln 2^3=3\ln 2=3\times 0.6931=2.0793\\ \ln 9=\ln 3^2=2\ln 3=2\times 1.0986=2.1972\\ \ln 10=\ln \left(2\times 5\right)=\ln 2+\ln 5=0.6931+1.6094=2.3025\\ \ln 12=\ln \left(2^2\times 3\right)=2\ln 2+\ln 3=2\times 0.6931+1.0986=2.4848\\ \ln 15=\ln \left(3\times 5\right)=\ln 3+\ln 5=1.0986+1.6094=2.708\\ \ln 16=\ln 2^4=4\ln 2=4\times 0.6931=2.7724\\ \ln 18=\ln \left(2\times 3^2\right)=\ln 2+2\ln 3=0.6931+2\times 1.0986=2.8903\end{array}$