Question

Find the value with the help of the logarithm tables:
\[5872\times 0.058\]

Answer 1

Hint: We have to find the product using the logarithm table. We will start by assuming \[x=5872\times 0.058\] and then apply a log on both sides. We will use the formulas such as \[\log \left( ab \right)=\log a+\log b,\log {{a}^{n}}=n\log a,\log 10=1\] to simplify. Once we get log x, we will apply an antilog to find the value of x.

Complete step by step answer:
We are given two numbers 5872 and 0.058. We have to find the product of them using the logarithm table. We will start by assuming our product \[5872\times 0.058\] as x. So, we can write it as
\[x=5872\times 0.058\]
Now, we will take log on both the sides, we will get,
\[\Rightarrow \log x=\log \left( 5872\times 0.058 \right)\]
We know that, \[\log ab=\log a+\log b,\] so using this in the above equation, we get,
\[\Rightarrow \log x=\log \left( 5872 \right)+\log \left( 0.058 \right)\]
Now, we will write our number in standard form for the logarithm table. So, we can express each of the two terms as shown below,
The first term is expressed as
\[5872=58.72\times 100=58.72\times {{10}^{2}}\]
And the second term can be expressed as
\[0.058=\dfrac{58}{1000}=58\times {{10}^{-3}}\]
Thus, we have,
\[\Rightarrow \log x=\log \left( 58.72\times {{10}^{2}} \right)+\log \left( 58\times {{10}^{-3}} \right)\]
Again using property \[\log ab=\log a+\log b,\] we get,
\[\Rightarrow \log x=\log \left( 58.72 \right)+\log \left( {{10}^{2}} \right)+\log \left( 58 \right)+\log \left( {{10}^{-3}} \right)\]
We also know that \[\log {{a}^{n}}=n\log a.\] So, we get,
\[\Rightarrow \log x=\log \left( 58.72 \right)+2\log 10+\log \left( 58 \right)-3\log 10\]
As we know the standard value of log 10 = 1, we get,
\[\Rightarrow \log x=\log \left( 58.72 \right)+2-3+\log \left( 58 \right)\]
Simplifying further, we get,
\[\Rightarrow \log x=-1+\log \left( 58.72 \right)+\log \left( 58 \right)\]
Using the table, we get,
\[\log 58.72=1.77\]
\[\log 58=1.76\]
So, we can substitute these in the original expression for x as
\[\Rightarrow \log x=-1+1.77+1.76\]
\[\Rightarrow \log x=2.53\]
Now, we have,
\[\Rightarrow \log x=2.53\]
Taking antilog on both the sides, we get,
\[\Rightarrow x=\text{Antilog}\left( 2.53 \right)\]
Using the table, we get,
\[\text{Antilog}\left( 2.53 \right)=338.84\]

So, we get,
\[x=338.84\]
\[\Rightarrow 5872\times 0.058=338.84\]

Note: Remember \[\log \left( ab \right)\ne \log a\log b\] and we have to solve using the correct formula, i.e. \[\log ab=\log a+\log b.\] Similarly, \[\log \left( \dfrac{a}{b} \right)\] is \[\log a-\log b.\] For verification, we can cross-check our answer by directly multiplying \[5872\times 0.058.\]