Question

Find the value of the given logarithmic term:
 $ {\text{lo}}{{\text{g}}_4}13.26 $ = ?

Answer 1

Hint: In order to find the value of the given logarithmic function, we observe that it cannot be directly solved, therefore we use a few logarithmic formulas to simplify and compute the given term. We make us of the identities,
 $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{M = lo}}{{\text{g}}_{\text{b}}}{\text{M}} \times {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b}} $
 $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{1}{{{\text{lo}}{{\text{g}}_{\text{b}}}{\text{a}}}} $

Complete step-by-step answer:
Given Data,
 $ {\text{lo}}{{\text{g}}_4}13.26 $
Using the formula of logarithmic terms, $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{M = lo}}{{\text{g}}_{\text{b}}}{\text{M}} \times {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b}} $
We can express $ {\text{lo}}{{\text{g}}_4}13.26 $ as:
 $ \Rightarrow {\text{lo}}{{\text{g}}_4}13.26{\text{ }} = {\text{ lo}}{{\text{g}}_{10}}13.26 \times {\text{lo}}{{\text{g}}_4}10 $
Now using the formula, $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{1}{{{\text{lo}}{{\text{g}}_{\text{b}}}{\text{a}}}} $ we can express the above equation in form of,
 $ \Rightarrow {\text{lo}}{{\text{g}}_4}13.26{\text{ }} = {\text{ lo}}{{\text{g}}_{10}}13.26 \times \dfrac{1}{{{\text{lo}}{{\text{g}}_{10}}4}} $
Using the logarithmic table we find the values of the terms,
 $ {\text{lo}}{{\text{g}}_{10}}13.26{\text{ = 1}}{\text{.1225}} $
 $ {\text{lo}}{{\text{g}}_{10}}4{\text{ = 0}}{\text{.6021}} $
Therefore we obtain, $ {\text{lo}}{{\text{g}}_4}13.26{\text{ = }}\dfrac{{1.1225}}{{0.6021}} $

Now let us consider some variable x such that, $ {\text{lo}}{{\text{g}}_4}13.26{\text{ = }}\dfrac{{1.1225}}{{0.6021}} = {\text{x}} $
Let us apply logarithm on both sides for this term, we get
 $ \Rightarrow {\text{log x = log}}\left( {\dfrac{{1.1225}}{{0.6021}}} \right) $
We know the formula, $ {\text{log a - log b = log}}\left( {\dfrac{{\text{a}}}{{\text{b}}}} \right) $
 $
   \Rightarrow {\text{log x = log 1}}{\text{.1225 - log 0}}{\text{.6021}} \\
   \Rightarrow {\text{log x = 0}}{\text{.0503 - }}\mathop 1\limits^\_ {\text{.7797}} \\
   \Rightarrow {\text{log x = 0}}{\text{.0503 - }}\left( { - 1 + 0.7797} \right) \\
   \Rightarrow {\text{log x = 0}}{\text{.0503 + 1 - 0}}{\text{.7797}} \\
   \Rightarrow {\text{log x = 0}}{\text{.2706}} \\
   \Rightarrow {\text{x = anti log 0}}{\text{.2706 = 1}}{\text{.865}} \\
 $

Hence the value of the given logarithmic term, $ {\text{lo}}{{\text{g}}_4}13.26 = 1.865 $

Note: In order to solve this type of problems the key is to know the concepts of logarithms and their relations. We are supposed to know the formulae of logs like, $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{M = lo}}{{\text{g}}_{\text{b}}}{\text{M}} \times {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b}} $ , $ {\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{1}{{{\text{lo}}{{\text{g}}_{\text{b}}}{\text{a}}}} $ and $ {\text{log a - log b = log}}\left( {\dfrac{{\text{a}}}{{\text{b}}}} \right) $ to be able to simplify the given terms.
Any natural logarithm is expressed with a base equal to 10 or ‘e’, where ‘e’ is a constant having the value approximately equal to 2.71.We find the value of log of a number or an anti-log of a number by referring to the logarithmic table, it has values to logarithms of almost all numbers including decimals.