Chapter 6, Problem 12CT

To determine

To describe : the expressions are equivalent

Answer to Problem 12CT

The expressions ${\log }_{5}11,\frac{\log 11}{\log 5},\frac{\ln 11}{\ln 5}$ are equivalent

Explanation of Solution

Given information The expressions are

  ${\log }_{5}11,\frac{\log 11}{\log 5},\text{ and }\frac{\ln 11}{\ln 5}$

  ${\log }_{5}11\text{ and }\frac{\log 11}{\log 5}$ are equivalent expression because the base of $\log$ replace in denominators by the change of base.

Therefore, ${\log }_{5}11=\frac{\log 11}{\log 5}$

  $\frac{\log 11}{\log 5}\text{ and }\frac{\ln 11}{\ln 5}$ are equivalent expression because the value of $\log$ and the value of $\ln$ is almost same.

Example:

  $\begin{array}{l}\log \sqrt{2}=0.151\\ \ln \sqrt{2}=0.346\end{array}$

Hence, the expressions ${\log }_{5}11,\frac{\log 11}{\log 5},\frac{\ln 11}{\ln 5}$ are equivalent.