Question

How do I find the approximate value of \[{\log _5}\left( {20} \right)\] ?

Answer 1

Hint: In order to solve this question, we will first change the base to \[10\] . For this we will use the change of base rule which is given by \[{\log _a}\left( x \right) = \dfrac{{{{\log }_b}\left( x \right)}}{{{{\log }_b}\left( a \right)}}\] where \[a\] and \[b\] are greater than \[0\] and not equal to \[1\] , and \[x\] is greater than \[0\] . After that we will substitute the values in the formula as \[a = 5,{\text{ }}x = 20\] and \[b = 10\] . Then we will find the values of \[{\log _{10}}\left( {20} \right)\] and \[{\log _{10}}\left( 5 \right)\] and simplify further to get the required approximate value of \[{\log _5}\left( {20} \right)\]

Complete step-by-step answer:
In the given problem, we are asked to find the approximate value of \[{\log _5}\left( {20} \right)\]
First of all, we will change the log base \[5\] to log base \[10\]
For this, we know that
According to the logarithm base change rule:
The base \[a\] logarithm of \[x\] is base \[b\] logarithm of \[x\] divided by the base \[b\] logarithm of \[a\]
i.e., \[{\log _a}\left( x \right) = \dfrac{{{{\log }_b}\left( x \right)}}{{{{\log }_b}\left( a \right)}}\]
where \[a\] and \[b\] are greater than \[0\] and not equal to \[1\] , and \[x\] is greater than \[0\] .
Now according to the given problem, we have
\[a = 5,{\text{ }}x = 20\] and \[b = 10\]
Therefore, on substituting the values in the formula, we have
\[{\log _5}\left( {20} \right) = \dfrac{{{{\log }_{10}}\left( {20} \right)}}{{{{\log }_{10}}\left( 5 \right)}}{\text{ }} - - - \left( i \right)\]
Now we will first the value of \[{\log _{10}}\left( {20} \right)\] and then find the value of \[{\log _{10}}\left( 5 \right)\]
Now, \[{\log _{10}}\left( {20} \right)\] can be written as
\[{\log _{10}}\left( {20} \right) = {\log _{10}}\left( {10 \times 2} \right)\]
As we know that
\[\log \left( {a \times b} \right) = \log \left( a \right) + \log \left( b \right)\]
Therefore, we have
\[{\log _{10}}\left( {10 \times 2} \right) = {\log _{10}}\left( {10} \right) + {\log _{10}}\left( 2 \right)\]
Now we know that
\[{\log _a}\left( a \right) = 1\]
\[ \Rightarrow {\log _{10}}\left( {10 \times 2} \right) = 1 + {\log _{10}}\left( 2 \right)\]
Since, \[{\log _{10}}\left( 2 \right) = 0.3010\]
Therefore, we get the value of \[{\log _{10}}\left( {20} \right)\] as
\[ \Rightarrow {\log _{10}}\left( {20} \right) = {\log _{10}}\left( {10 \times 2} \right) = 1 + 0.3010\]
\[ \Rightarrow {\log _{10}}\left( {20} \right) = 1.3010\]
Now, we will find the value of \[{\log _{10}}\left( 5 \right)\]
Now, \[{\log _{10}}\left( 5 \right)\] can be written as
\[{\log _{10}}\left( 5 \right) = {\log _{10}}\left( {\dfrac{{10}}{2}} \right)\]
As we know that
\[\log \left( {\dfrac{a}{b}} \right) = \log \left( a \right) - \log \left( b \right)\]
Therefore, we have
\[{\log _{10}}\left( 5 \right) = {\log _{10}}\left( {10} \right) - {\log _{10}}\left( 2 \right)\]
Now we know that
\[{\log _a}\left( a \right) = 1\]
\[ \Rightarrow {\log _{10}}\left( 5 \right) = 1 - {\log _{10}}\left( 2 \right)\]
Since, \[{\log _{10}}\left( 2 \right) = 0.3010\]
Therefore, we get the value of \[{\log _{10}}\left( 5 \right)\] as
\[ \Rightarrow {\log _{10}}\left( 5 \right) = 1 - 0.3010\]
\[ \Rightarrow {\log _{10}}\left( 5 \right) = 0.6989\]
Now on substituting the values in the equation \[\left( i \right)\] we get
\[ \Rightarrow {\log _5}\left( {20} \right) = \dfrac{{{{\log }_{10}}\left( {20} \right)}}{{{{\log }_{10}}\left( 5 \right)}} = \dfrac{{1.3010}}{{0.6989}}\]
On simplifying, we get
\[ \Rightarrow {\log _5}\left( {20} \right) \approx 1.86149\]
which is the required result.
Hence, the approximate value of \[{\log _5}\left( {20} \right)\] is \[1.86149\]

Note: While solving these type of problems, you should always first try to change the base to \[10\] You can also calculate the values by looking at the log table and then directly writing the answer, which is a shortcut method, but if the log table is not available then you should keep in mind the basic properties used to solve the logarithmic problem. Also, you should know the value of \[{\log _{10}}\left( 2 \right)\] as it is basic to find the other values. Also, while using the base change rule, don’t get confused between the values of \[a\] and \[b\] as it can cause errors while solving further. So, apply it properly and simplify the values carefully.