Question

How do you estimate the value of ${{\log }_{10}}6.25$?

Answer 1

Hint: Assume the given expression as ‘E’. Now, first simplify the argument of the log by converting it into the fractional form. In the next step use the property of log given as $\log \left( mn \right)=\log m+\log n$. Now, write the argument of the two log terms in their exponential form and use the formula $\log {{a}^{m}}=m\log a$ for further simplification. Use the value: ${{\log }_{10}}2=0.3010$ to get the answer.

Complete step by step solution:
Here, we have been provided with the logarithmic expression ${{\log }_{10}}6.25$ and we are asked to estimate its value.
Now, let us assume the value of the given expression as E. So, we have,
$\Rightarrow E={{\log }_{10}}6.25$
Now, first we need to convert the argument of the log which is 6.25 into the fractional form, so we have,
$\Rightarrow E={{\log }_{10}}\left( \dfrac{625}{100} \right)$
Using the property of log given as $\log \left( mn \right)=\log m+\log n$, we get,
$\Rightarrow E={{\log }_{10}}625-{{\log }_{10}}100$
Converting the argument of the two log expressions into their exponential form, we get,
$\Rightarrow E={{\log }_{10}}\left( {{5}^{4}} \right)-{{\log }_{10}}\left( {{10}^{2}} \right)$
Using the formula: $\log {{a}^{m}}=m\log a$, we get,
$\begin{align}
  & \Rightarrow E=4{{\log }_{10}}5-2{{\log }_{10}}10 \\
 & \Rightarrow E=4{{\log }_{10}}5-2\times 1 \\
 & \Rightarrow E=4{{\log }_{10}}5-2 \\
\end{align}$
Now, the above expression can be written as:
$\begin{align}
  & \Rightarrow E=4{{\log }_{10}}\left( \dfrac{10}{2} \right)-2 \\
 & \Rightarrow E=4\left( {{\log }_{10}}10-{{\log }_{10}}2 \right)-2 \\
 & \Rightarrow E=4\left( 1-{{\log }_{10}}2 \right)-2 \\
 & \Rightarrow E=4-4{{\log }_{10}}2-2 \\
 & \Rightarrow E=2-4{{\log }_{10}}2 \\
\end{align}$
Substituting the value ${{\log }_{10}}2=0.3010$, we get,
$\begin{align}
  & \Rightarrow E=2-4\left( 0.3010 \right) \\
 & \Rightarrow E=2-1.2040 \\
 & \Rightarrow E=0.7960 \\
\end{align}$
Hence, the estimated value of the given logarithmic expression is 0.7960.

Note: One must remember the log values of arguments from 1 to 10 with the base value of the log equal to 10. You may use the log table to substitute the value ${{\log }_{10}}2=0.3010$, however you need to learn how to calculate the log values using the log table. The values of $\log 2,\log 3,\log 5,\log 7$ must be remembered because many times you may not be provided with the log table or values in the question. Also remember the relation between the natural log and the common log given as $\ln a=2.303\log a$.