Chapter 3.3, Problem 97E

To determine

To rewrite: logarithm $f\left(x\right)={\log }_{2}x$ as a ratio of logarithms and graph the ratio thus obtained.

Answer to Problem 97E

  ${\log }_{2}x=\frac{\log x}{\log 2}=\frac{\ln x}{\ln 2}$

Explanation of Solution

Concept Used:

Change-of-Base Formula: Let a , b , and x be positive real numbers such that $a\ne 1$ and $b\ne 1$ . Then ${\log }_{a}x$ can be converted to a different base as follows.

Base b : ${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$

Base 10: ${\log }_{a}x=\frac{\log x}{\log a}$

Base e : ${\log }_{a}x=\frac{\ln x}{\ln a}$

Consider the logarithm $f\left(x\right)={\log }_{2}x$ , using the change of base formula it can be written as:

  ${\log }_{2}x=\frac{\log x}{\log 2}=\frac{\ln x}{\ln 2}$

The graph of the ratio obtained above is: