Question

How do you use the Change of Base Formula and a calculator to evaluate the logarithms \[{\log _3}21.8\]\[?\]

Answer 1

Hint: Given a logarithm of the form \[{\log _b}M\], use the change-of-base formula to rewrite it as a quotient of logs with any positive base \[n\], where \[n \ne 1\]. determine the new base \[n\], remembering that the common \[\log \], \[\log \left( x \right)\], has base 10 and the natural \[\log \], \[\ln (x)\], has base \[e\]. Rewrite the \[\log \] as a quotient using the change-of-base formula:

Complete step by step answer:
The change-of-base formula can be used to evaluate a logarithm with any base.
Given a logarithm in the question of the form \[{\log _b}M\], by use the change of base formula the given logarithm function can be rewrite as a quotient of logs with any positive real numbers M, b, and n, where \[n \ne 1\] and \[b \ne 1\], as follows
The numerator of the quotient will be a logarithm with base n and argument M and the denominator of the quotient will be a logarithm with base n and argument b.
By this, the change-of-base formula can be used to rewrite a logarithm with base n as the quotient of common or natural logs.
\[{\log _b}M = \dfrac{{\ln M}}{{\ln b}}\] and \[{\log _b}M = \dfrac{{{{\log }_n}M}}{{{{\log }_n}b}}\]
Remember the standard base value n for common \[\log \], \[\log \left( x \right)\]has base value 10 and the natural \[\log \], \[\ln (x)\] has base \[e\].
Now to evaluate the given common logarithms \[{\log _3}21.8\] by use the change of base formula is
For common logarithm as we know the value of new base n is 10.
\[ \Rightarrow \,\,{\log _3}21.8 = \dfrac{{{{\log }_{10}}21.8}}{{{{\log }_{10}}3}}\]
By using a logarithm calculator with base 10 the value of \[{\log _{10}}21.8 = 1.33845649\] and \[{\log _{10}}3 = 0.477121255\].
\[ \Rightarrow \,\,{\log _3}21.8 = \dfrac{{1.33845649}}{{0.477121255}}\]
\[\therefore \,\,\,\,{\log _3}21.8 = 2.8052753382\]

Therefore, by use the Change of Base Formula and a calculator the value of logarithms \[{\log _3}21.8\] is \[2.8052753382\].

Note: The logarithmic function is a reciprocal or the inverse of exponential function. To solve the question, we must know about the properties of the logarithmic function. There are properties on addition, subtraction, product, division etc., on the logarithmic functions. We have to change the base of the log function and to simplify the given question.