Question

How do you find the exact value of \[{\log _2}\left( { - 16} \right)\]?

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 and further simplify by using logarithm properties, to get the required solution.

Complete step by step solution:
Consider a given logarithm function
\[ \Rightarrow {\log _2}\left( { - 16} \right)\]
Now, we have to find the exact value of the function, this can be solve by using a change of base formula.
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 \[{\log _2}\left( { - 16} \right)\] by use the change of base formula is
\[ \Rightarrow {\log _2}\left( { - 16} \right) = \dfrac{{\ln \left( { - 16} \right)}}{{\ln 2}}\]
As we know the 16 id the 4th root of 2 and value of \[{i^2} = - 1\], then \[ - 16 = {2^4}\]
Therefore,
\[ \Rightarrow \dfrac{{\ln \left( {\left( {{i^2}} \right){2^4}} \right)}}{{\ln 2}}\]
Apply logarithm properties\[\ln \left( {ab} \right) = \ln a + \ln b\] in numerator, then
\[ \Rightarrow \dfrac{{\ln \left( {{i^2}} \right) + \ln \left( {{2^4}} \right)}}{{\ln 2}}\]
Again, apply logarithm properties \[\ln \left( {{a^n}} \right) = n\ln a\] in numerator, then
\[ \Rightarrow \dfrac{{2\ln \left( i \right) + 4\ln \left( 2 \right)}}{{\ln 2}}\]
Separate the fraction as
\[ \Rightarrow \dfrac{{2\ln \left( i \right)}}{{\ln 2}} + \,\dfrac{{4\ln \left( 2 \right)}}{{\ln 2}}\]
On simplification, we get
\[ \Rightarrow \dfrac{{2\ln \left( i \right)}}{{\ln 2}} + 4\]
As we know the value of \[\ln \left( i \right) = i\dfrac{\pi }{2}\], on substituting we have
\[ \Rightarrow \dfrac{{2\left( {i\dfrac{\pi }{2}} \right)}}{{\ln 2}} + 4\]
On simplification, we get
\[ \Rightarrow \dfrac{{\pi i}}{{\ln 2}} + 4\]
And by using logarithm calculator the value of \[\ln 2 = 0.6931471806\] and the standard value \[\pi = 3.1428571429\], then
\[ \Rightarrow \dfrac{{3.1428571429i}}{{0.6931471806}} + 4\]
\[ \Rightarrow \dfrac{{3.1428571429i}}{{0.6931471806}} + 4\]
\[ \Rightarrow 4.534184414i + 4\]
\[ \Rightarrow 4 + 4.534184414\,i\]
Hence, the value of \[{\log _2}\left( { - 16} \right) = \,4 + 4.534184414\,i\].
So, the correct answer is “\[\,4 + 4.534184414\,i\]”.

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.