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.